{"id":35220,"date":"2022-08-25T11:30:27","date_gmt":"2022-08-25T06:00:27","guid":{"rendered":"https:\/\/apboardsolutions.guru\/?p=35220"},"modified":"2022-08-25T11:30:27","modified_gmt":"2022-08-25T06:00:27","slug":"inter-2nd-year-maths-2b-circle-formulas","status":"publish","type":"post","link":"https:\/\/apboardsolutions.guru\/inter-2nd-year-maths-2b-circle-formulas\/","title":{"rendered":"Inter 2nd Year Maths 2B Circle Formulas"},"content":{"rendered":"<p>Use these <a href=\"https:\/\/apboardsolutions.guru\/inter-2nd-year-maths-2b-formulas\/\">Inter 2nd Year Maths 2B Formulas<\/a> PDF Chapter 1 Circle to solve questions creatively.<\/p>\n<h2>Intermediate 2nd Year Maths 2B Circle Formulas<\/h2>\n<p>\u2192 The locus of a point in a plane such that its distance from a fixed point in the plane is always the same is called a circle.<\/p>\n<p>\u2192 The equation of a circle with centre (h, k) and radius r is (x &#8211; h)<sup>2<\/sup> + (y &#8211; k)<sup>2<\/sup> = r<sup>2<\/sup><\/p>\n<p>\u2192 The equation of a circle in standard form is x<sup>2<\/sup> + y<sup>2<\/sup> = r<sup>2<\/sup>.<\/p>\n<p>\u2192 The equation of a circle in general form is x<sup>2<\/sup> + y<sup>2<\/sup> + 2gx + 2fy + c = 0 and its centre is (-g, -f), radius is \\(\\sqrt{g^{2}+f^{2}-c}\\).<\/p>\n<p><img loading=\"lazy\" src=\"https:\/\/apboardsolutions.com\/wp-content\/uploads\/2022\/07\/AP-Board-Solutions.png\" alt=\"Inter 2nd Year Maths 2B Circle Formulas\" width=\"196\" height=\"17\" \/><\/p>\n<p>\u2192 The intercept made by x<sup>2<\/sup> + y<sup>2<\/sup> + 2gx + 2fy + c = 0<\/p>\n<ul>\n<li>on X-axis is 2\\(\\sqrt{g^{2}-c}\\) if g<sup>2<\/sup> &gt; c.<\/li>\n<li>on Y-axis is 2\\(\\sqrt{f^{2}-c}\\) if f<sup>2<\/sup> &gt; c.<\/li>\n<\/ul>\n<p>\u2192 If the extremities of a diameter of a circle are (x<sub>1<\/sub>, y<sub>1<\/sub>) and (x<sub>2<\/sub>, y<sub>2<\/sub>) then its equation is (x &#8211; x<sub>1<\/sub>) (x &#8211; x<sub>2<\/sub>) + (y &#8211; y<sub>1<\/sub>) (y &#8211; y<sub>2<\/sub>) = 0<\/p>\n<p>\u2192 The equation of a circle passing through three non-collinear points (x<sub>1<\/sub>, y<sub>1<\/sub>), (x<sub>2<\/sub>, y<sub>2<\/sub>) and (x<sub>3<\/sub>, y<sub>3<\/sub>) is<br \/>\n\\(\\left|\\begin{array}{lll}<br \/>\nx_{1} &amp; y_{1} &amp; 1 \\\\<br \/>\nx_{2} &amp; y_{2} &amp; 1 \\\\<br \/>\nx_{3} &amp; y_{3} &amp; 1<br \/>\n\\end{array}\\right|\\) = (x<sup>2<\/sup> + y<sup>2<\/sup>) + \\(\\left|\\begin{array}{lll}<br \/>\nc_{1} &amp; y_{1} &amp; 1 \\\\<br \/>\nc_{2} &amp; y_{2} &amp; 1 \\\\<br \/>\nc_{3} &amp; y_{3} &amp; 1<br \/>\n\\end{array}\\right|\\) x + \\(\\left|\\begin{array}{lll}<br \/>\nx_{1} &amp; C_{1} &amp; 1 \\\\<br \/>\nx_{2} &amp; C_{2} &amp; 1 \\\\<br \/>\nx_{3} &amp; C_{3} &amp; 1<br \/>\n\\end{array}\\right|\\) y + \\(\\left|\\begin{array}{lll}<br \/>\nx_{1} &amp; y_{1} &amp; C_{1} \\\\<br \/>\nx_{2} &amp; y_{2} &amp; C_{2} \\\\<br \/>\nx_{3} &amp; y_{3} &amp; C_{3}<br \/>\n\\end{array}\\right|\\) = 0.<br \/>\nwhere c<sub>i<\/sub> = &#8211; (x<sub>i<\/sub><sup>2<\/sup> + y<sub>i<\/sub><sup>2<\/sup>)<\/p>\n<p>\u2192 The centre of the circle passing through three non-collinear points (x<sub>1<\/sub>, y<sub>1<\/sub>), (x<sub>2<\/sub>, y<sub>2<\/sub>) and (x<sub>3<\/sub>, y<sub>3<\/sub>) is<br \/>\n\\(\\left[\\frac{\\left|\\begin{array}{lll}<br \/>\nc_{1} &amp; y_{1} &amp; 1 \\\\<br \/>\nc_{2} &amp; y_{2} &amp; 1 \\\\<br \/>\nc_{3} &amp; y_{3} &amp; 1<br \/>\n\\end{array}\\right|}{(-2)\\left|\\begin{array}{lll}<br \/>\nx_{1} &amp; y_{1} &amp; 1 \\\\<br \/>\nx_{2} &amp; y_{2} &amp; 1 \\\\<br \/>\nx_{3} &amp; y_{3} &amp; 1<br \/>\n\\end{array}\\right|}, \\frac{\\left|\\begin{array}{lll}<br \/>\nx_{1} &amp; c_{1} &amp; 1 \\\\<br \/>\nx_{2} &amp; c_{2} &amp; 1 \\\\<br \/>\nx_{3} &amp; c_{3} &amp; 1<br \/>\n\\end{array}\\right|}{(-2)\\left|\\begin{array}{lll}<br \/>\nx_{1} &amp; y_{1} &amp; 1 \\\\<br \/>\nx_{2} &amp; y_{2} &amp; 1 \\\\<br \/>\nx_{3} &amp; y_{3} &amp; 1<br \/>\n\\end{array}\\right|}\\right]\\)<\/p>\n<p><img loading=\"lazy\" src=\"https:\/\/apboardsolutions.com\/wp-content\/uploads\/2022\/07\/AP-Board-Solutions.png\" alt=\"Inter 2nd Year Maths 2B Circle Formulas\" width=\"196\" height=\"17\" \/><\/p>\n<p>\u2192 The parametric equations of a circle with centre (h, k) and radius (r \u2265 0) are given by<br \/>\nx = h + r cos \u03b8<br \/>\ny = k + r sin \u03b8 0 \u2264 6 &lt; 2\u03c0.<\/p>\n<p>\u2192 A point P(x<sub>1<\/sub>, y<sub>1<\/sub>) is an interior point or on the circumference or an exterior point of a circles S = 0 \u21d4 S<sub>11<\/sub> \\(\\frac{&lt;}{&gt;}\\) 0.<\/p>\n<p>\u2192 The power of P(x<sub>1<\/sub>, y<sub>1<\/sub>) with respect to the circle S = 0 is S<sub>11<\/sub>.<\/p>\n<p>\u2192 A point P(x<sub>1<\/sub>, y<sub>1<\/sub>) is an interior point or on the circumference or exterior point of the circle S = 0 \u21d4 the power of P with respect to S = 0 is negative, zero and positive.<\/p>\n<p>\u2192 If a straight line through a point P(x<sub>1<\/sub>, y<sub>1<\/sub>) meets the circle S = 0 at A and B then the power of P is equal to PA. PB.<\/p>\n<p>\u2192 The length of the tangent from P(x<sub>1<\/sub>, y<sub>1<\/sub>) to S = 0 is \\(\\sqrt{S_{11}}\\).<\/p>\n<p>\u2192 The straight line l = 0 intersects, touches or does not meet the circles = 0 according as l &lt; r, l = r or l &gt; r where l is the perpendicular distance from the centre of the circle to the line l = 0 and r is the radius.<\/p>\n<p>\u2192 For every real value of m the straight line y = mx \u00b1 r \\(\\sqrt{1+m^{2}}\\) is a tangent to the circle x<sup>2<\/sup> + y<sup>2<\/sup> = r<sup>2<\/sup>.<\/p>\n<p>\u2192 If r is the radius of the circle S = x<sup>2<\/sup> + y<sup>2<\/sup> + 2gx + 2fy + c = 0 then for every real value of m the straight line y + f = m(n + g) \u00b1r + m2 will be a tangent to the circle.<\/p>\n<p>\u2192 If P(x<sub>1<\/sub>, y<sub>1<\/sub>) and Q(x<sub>2<\/sub>, y<sub>2<\/sub>) are two points on the circle S = 0 then the secant&#8217;s \\((\\stackrel{\\leftrightarrow}{P Q})\\) equation is S<sub>1<\/sub> + S<sub>2<\/sub> = S<sub>12<\/sub><\/p>\n<p>\u2192 The equation of tangent at (x<sub>1<\/sub>, y<sub>1<\/sub>) of the circle S = 0 is S<sub>1<\/sub> = 0.<\/p>\n<p>\u2192 If \u03b8<sub>1<\/sub>, \u03b8<sub>2<\/sub> are two points on S = x<sup>2<\/sup> + y<sup>2<\/sup> + 2gx + 2fy + c = 0 then the equation of the chord joining the points \u03b8<sub>1<\/sub>, \u03b8<sub>2<\/sub> is<br \/>\n(x + g) cos \\(\\left(\\frac{\\theta_{1}+\\theta_{2}}{2}\\right)\\) + (y + f) sin \\(\\left(\\frac{\\theta_{1}+\\theta_{2}}{2}\\right)\\) = r cos \\(\\left(\\frac{\\theta_{1}-\\theta_{2}}{2}\\right)\\)<\/p>\n<p>\u2192 The equation of the tangent at \u03b8 of the circle S = 0 is (x + g) cos \u03b8 + (y + f) sin \u03b8 = r.<\/p>\n<p>\u2192 The equation of normal at (x<sub>1<\/sub>, y<sub>1<\/sub>) of the circle<br \/>\nS = 0 is (x &#8211; x<sub>1<\/sub>) (y<sub>1<\/sub> + f) &#8211; (y &#8211; y<sub>1<\/sub>) (x<sub>1<\/sub> + g) = 0.<\/p>\n<p><img loading=\"lazy\" src=\"https:\/\/apboardsolutions.com\/wp-content\/uploads\/2022\/07\/AP-Board-Solutions.png\" alt=\"Inter 2nd Year Maths 2B Circle Formulas\" width=\"196\" height=\"17\" \/><\/p>\n<p>\u2192 The chord of contact of P(x<sub>1<\/sub> y<sub>1<\/sub>) (exterior point) with respect to S = 0 is S<sub>1<\/sub> = 0.<\/p>\n<p>\u2192 The equation of the polar of a point P(x<sub>1<\/sub>, y<sub>1<\/sub>) with respect to S = 0 is S<sub>1<\/sub> = 0.<\/p>\n<table border=\"2\">\n<tbody>\n<tr>\n<td>P(x<sub>1<\/sub>, y<sub>1<\/sub>)<\/td>\n<td>Tangent at P<\/td>\n<td>Chord of contact at p<\/td>\n<td>Polar of P<\/td>\n<\/tr>\n<tr>\n<td>(i) Interior of the circle<\/td>\n<td>Does not exist<\/td>\n<td>Does not exist<br \/>\n(not defined)<\/td>\n<td>S<sub>1<\/sub> = 0<br \/>\n(P is different from the centre of the circle)<\/td>\n<\/tr>\n<tr>\n<td>(ii) On the circle<\/td>\n<td>S<sub>1<\/sub> = 0<\/td>\n<td>S<sub>1<\/sub> = 0<\/td>\n<td>S<sub>1<\/sub> = 0<\/td>\n<\/tr>\n<tr>\n<td>(iii) Exterior of the circle<\/td>\n<td>Does not exist<\/td>\n<td>S<sub>1<\/sub> = 0<\/td>\n<td>S<sub>1<\/sub> = 0<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>\u2192 The pole of lx + my + n = 0 with respect to S = 0 is<br \/>\n\\(\\left(-g+\\frac{l r^{2}}{l g+m f-n},-f+\\frac{m r^{2}}{l g+m f-n}\\right)\\)<\/p>\n<p>\u2192 Where r is the radius of the circle. The polar of P(x<sub>1<\/sub>, y<sub>1<\/sub>) with respect to S = 0 passes through Q(x<sub>2<\/sub>, y<sub>2<\/sub>) \u21d4 the polar of Q with respect to S &#8211; 0 passes through P.<\/p>\n<p>\u2192 The points (x<sub>1<\/sub>, y<sub>1<\/sub>) and (x<sub>2<\/sub>, y<sub>2<\/sub>) are conjugate points with respect to S = 0 if S<sub>12<\/sub> = 0<\/p>\n<p>\u2192 Two lines l<sub>1<\/sub>x = m<sub>1<\/sub>y + n<sub>1<\/sub> = 0, l<sub>2<\/sub>x + m<sub>2<\/sub>y + n<sub>2<\/sub> = 0 are conjugate with respect to x<sup>2<\/sup> + y<sup>2<\/sup> = a<sup>2<\/sup> \u21d4 (l<sub>1<\/sub>l<sub>2<\/sub> + m<sub>1<\/sub>m<sub>2<\/sub>) = n<sub>1<\/sub>n<sub>2<\/sub><\/p>\n<p>\u2192 Two points P, Q are said to be inverse points with respect to S = 0 if CP. CQ = r<sup>2<\/sup> where C is the centre and r is the radius of the circle S = 0.<\/p>\n<p>\u2192 If (x<sub>1<\/sub>, y<sub>1<\/sub>) is the mid-point of a chord of the circle S = 0 then its chord equation is S<sub>1<\/sub> = S<sub>11<\/sub>.<\/p>\n<p>\u2192 The pair of common tangents to the circles S = 0, S&#8217; = 0 touching at a point on the lines segment \\(\\overline{\\mathrm{C}_{1} \\mathrm{C}_{2}}\\) (C<sub>1<\/sub>, C<sub>2<\/sub> are centres of the circles) is called transverse pair of common tangents.<\/p>\n<p>\u2192 The pair of common tangents to the circles S = 0, S&#8217; = 0 intersecting at a point not in \\(\\overline{\\mathrm{C}_{1} \\mathrm{C}_{2}}\\) is called as direct pair of common tangents.<\/p>\n<p><img loading=\"lazy\" src=\"https:\/\/apboardsolutions.com\/wp-content\/uploads\/2022\/07\/AP-Board-Solutions.png\" alt=\"Inter 2nd Year Maths 2B Circle Formulas\" width=\"196\" height=\"17\" \/><\/p>\n<p>\u2192 The point of intersection of transvese (direct) common tangents is called internal (external) Centre of similitude.<\/p>\n<table border=\"2\">\n<tbody>\n<tr>\n<td width=\"208\">Situation<\/td>\n<td width=\"208\">No of common tangents<\/td>\n<\/tr>\n<tr>\n<td width=\"208\">1. \\( \\overline{C_{1} C_{2}} \\) &gt; r<sub>1<\/sub> + r<sub>2<\/sub><\/td>\n<td width=\"208\">4<\/td>\n<\/tr>\n<tr>\n<td width=\"208\">2. r<sub>1<\/sub> + r<sub>2<\/sub> = \\( \\overline{C_{1} C_{2}} \\)<\/td>\n<td width=\"208\">3<\/td>\n<\/tr>\n<tr>\n<td width=\"208\">3. |r<sub>1<\/sub> \u2013 r<sub>2<\/sub>| &lt; \\( \\overline{C_{1} C_{2}} \\) &lt; r<sub>1<\/sub> + r<sub>2<\/sub><\/td>\n<td width=\"208\">2<\/td>\n<\/tr>\n<tr>\n<td width=\"208\">4. C<sub>1<\/sub>C<sub>2<\/sub> = |r<sub>1<\/sub> \u2013 r<sub>2<\/sub>|<\/td>\n<td width=\"208\">1<\/td>\n<\/tr>\n<tr>\n<td width=\"208\">5. C<sub>1<\/sub>C<sub>2<\/sub> &lt; |r<sub>1<\/sub> \u2013 r<sub>2<\/sub>|<\/td>\n<td width=\"208\">0<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>\u2192 The combined equation of the pair of tangents drawn from an external point P(x<sub>1<\/sub>, y<sub>1<\/sub>) to the circle S = 0 is SS<sub>11<\/sub> = S<sup>2<\/sup><sub>1<\/sub>.<\/p>\n<p>Equation of a Circle:<br \/>\nThe equation of the circle with centre C (h, k) and radius r is (x &#8211; h)<sup>2<\/sup> + (y &#8211; k)<sup>2<\/sup> = r<sup>2<\/sup>.<br \/>\nProof:<br \/>\nLet P(x<sub>1<\/sub>, y<sub>1<\/sub>) be a point on the circle.<br \/>\nP lies in the circle \u21d4 PC = r \u21d4 \\(\\sqrt{\\left(\\mathrm{x}_{1}-\\mathrm{h}\\right)^{2}+\\left(\\mathrm{y}_{1}-\\mathrm{k}\\right)^{2}}\\) = r<br \/>\n\u21d4 (x<sub>1<\/sub> &#8211; h)<sup>2<\/sup> + (y<sub>1<\/sub> &#8211; k)<sup>2<\/sup> = r<sup>2<\/sup>.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-20454\" src=\"https:\/\/apboardsolutions.com\/wp-content\/uploads\/2022\/08\/Inter-2nd-Year-Maths-2B-Circle-Formulas-1.png\" alt=\"Inter 2nd Year Maths 2B Circle Formulas 1\" width=\"294\" height=\"122\" \/><br \/>\nThe locus of P is (x &#8211; h)<sup>2<\/sup> + (y &#8211; k)<sup>2<\/sup> = r<sup>2<\/sup>.<br \/>\n\u2234 The equation of the circle is (x &#8211; h)<sup>2<\/sup> + (y &#8211; k)<sup>2<\/sup> = r<sup>2<\/sup>.<\/p>\n<p>Note: The equation of a circle with centre origin and radius r is (x &#8211; 0)<sup>2<\/sup> + (y &#8211; 0)<sup>2<\/sup> = r<sup>2<\/sup><br \/>\ni.e., x<sup>2<\/sup> + y<sup>2<\/sup> = r<sup>2<\/sup> which is the standard equation of the circle.<\/p>\n<p>Note: On expanding equation (1), the equation of a circle is of the form x<sup>2<\/sup> + y<sup>2<\/sup> + 2gx + 2fy + c = 0.<\/p>\n<p>Theorem: If g<sup>2<\/sup> + f<sup>2<\/sup> &#8211; c \u2265 0, then the equation x<sup>2<\/sup> + y<sup>2<\/sup> + 2gx + 2fy + c = 0 represents a circle with centre (- g, &#8211; f) and radius \\(\\sqrt{g^{2}+f^{2}-c}\\).<br \/>\nNote: If ax<sup>2<\/sup> + ay<sup>2<\/sup> + 2gx + 2fy + c = 0 represents a circle, then its centre = \\(\\left(-\\frac{g}{a},-\\frac{f}{a}\\right)\\) and its radius \\(\\frac{\\sqrt{\\mathrm{g}^{2}+\\mathrm{f}^{2}-\\mathrm{ac}}}{|\\mathrm{a}|}\\).<\/p>\n<p>Theorem: The equation of a circle having the line segment joining A(x<sub>1<\/sub>, y<sub>1<\/sub>) and B(x<sub>2<\/sub>, y<sub>2<\/sub>) as diameter is (x &#8211; x<sub>1<\/sub>) (x &#8211; x<sub>2<\/sub>) + (y &#8211; y<sub>1<\/sub>) (y &#8211; y<sub>2<\/sub>) = 0.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-20453\" src=\"https:\/\/apboardsolutions.com\/wp-content\/uploads\/2022\/08\/Inter-2nd-Year-Maths-2B-Circle-Formulas-2.png\" alt=\"Inter 2nd Year Maths 2B Circle Formulas 2\" width=\"355\" height=\"155\" \/><br \/>\nLet P(x, y) be any point on the circle. Given points A(x<sub>1<\/sub>, y<sub>1<\/sub>) and B(x<sub>2<\/sub>, y<sub>2<\/sub>).<br \/>\nNow \u2220APB = \\(\\frac{\\pi}{2}\\). (Angle in a semi circle.)<br \/>\nSlope of AP. Slope of BP = &#8211; 1<br \/>\n\u21d2 \\(\\frac{y-y_{1}}{x-x_{1}} \\frac{y-y_{2}}{x-x_{2}}\\) = &#8211; 1<br \/>\n\u21d2 (y &#8211; y<sub>2<\/sub>) (y &#8211; y<sub>1<\/sub>) = &#8211; (x &#8211; x<sub>2<\/sub>) (x &#8211; x<sub>1<\/sub>) = 0<br \/>\n\u21d2 (x &#8211; x<sub>2<\/sub>) (x &#8211; x<sub>1<\/sub>) + (y &#8211; y<sub>2<\/sub>) (y &#8211; y<sub>1<\/sub>) = 0<\/p>\n<p>Definition: Two circles are said to be concentric if they have same center.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-20452\" src=\"https:\/\/apboardsolutions.com\/wp-content\/uploads\/2022\/08\/Inter-2nd-Year-Maths-2B-Circle-Formulas-3.png\" alt=\"Inter 2nd Year Maths 2B Circle Formulas 3\" width=\"308\" height=\"116\" \/><br \/>\nThe equation of the circle concentric with the circle x<sup>2<\/sup> + y<sup>2<\/sup> + 2gx + 2fy + c = 0 is of the form x<sup>2<\/sup> + y<sup>2<\/sup> + 2gx + 2fy + k = 0.<br \/>\nThe equation of the concentric circles differs by constant only.<\/p>\n<p>Parametric Equations of A Circle:<\/p>\n<p>Theorem: If P(x, y) is a point on the circle with centre C(\u03b1, \u03b2) and radius r, then x = \u03b1 + r cos\u03b8, y = \u03b2 + r sin \u03b8 where 0 \u2264 \u03b8 &lt; 2\u03c0.<\/p>\n<p>Note: The equations x = \u03b1 + r cos \u03b8, y = + r sin \u03b8, 0 \u2264 \u03b8 &lt; 2\u03c0 are called parametric equations of the circle with centre (\u03b1, \u03b2) and radius r.<\/p>\n<p>Note: A point on the circle x<sup>2<\/sup> + y<sup>2<\/sup> = r<sup>2<\/sup> is taken in the form (r cos \u03b8, r sin \u03b8). The point (r cos \u03b8, r sin \u03b8) is simply denoted as point \u03b8.<\/p>\n<p>Theorem:<br \/>\n(1) If g<sup>2<\/sup> &#8211; c &gt; 0 then the intercept made on the x axis by the circle x<sup>2<\/sup> + y<sup>2<\/sup> + 2gx + 2fy + c = 0 is 2\\(\\sqrt{g^{2}-a c}\\)<br \/>\n(2) If f<sup>2<\/sup> &#8211; c &gt;0 then the intercept made on the y axis by the circle x<sup>2<\/sup> + y<sup>2<\/sup> + 2gx + 2fy + c = 0 is 2\\(\\sqrt{f^{2}-b c}\\)<\/p>\n<p>Note: The condition for the x-axis to touch the circle<br \/>\nx<sup>2<\/sup> + y<sup>2<\/sup> + 2gx + 2fy + c = 0 (c &gt; 0) is g<sup>2<\/sup> = c.<\/p>\n<p>Note: The condition of the y-axis to touch the circle<br \/>\nx<sup>2<\/sup> + y<sup>2<\/sup> + 2gx + 2fy + c = 0 (c &gt; 0) is f<sup>2<\/sup> = c.<\/p>\n<p>Position of Point:<br \/>\nLet S = 0 be a circle and P(x<sub>1<\/sub>, y<sub>1<\/sub>) be a point I in the plane of the circle. Then<\/p>\n<ul>\n<li>P lies inside the circle S = 0 \u21d4 S<sub>11<\/sub> &lt; 0<\/li>\n<li>P lies in the circle S = 0 \u21d4 S<sub>11<\/sub> = 0<\/li>\n<li>Plies outside the circle S = 0 \u21d4 S<sub>11<\/sub> = 0<\/li>\n<\/ul>\n<p>Power of a Point:<br \/>\nLet S = 0 be a circle with centre C and radius r. Let P be a point. Then CP<sup>2<\/sup> &#8211; r<sup>2<\/sup> is called power of P with respect to the circle S = 0.<\/p>\n<p>Theorem: The power of a point P(x<sub>1<\/sub>, y<sub>1<\/sub>) with respect to the circle S = 0 is S<sub>11<\/sub>.<\/p>\n<p>Theorem: The length of the tangent drawn from an external point P(x<sub>1<\/sub>, y<sub>1<\/sub>) to the circle s = 0 is \\(\\sqrt{\\mathrm{S}_{11}}\\).<\/p>\n<p><img loading=\"lazy\" src=\"https:\/\/apboardsolutions.com\/wp-content\/uploads\/2022\/07\/AP-Board-Solutions.png\" alt=\"Inter 2nd Year Maths 2B Circle Formulas\" width=\"196\" height=\"17\" \/><\/p>\n<p>Theorem: The equation of the tangent to the circle S = 0 at P(x<sub>1<\/sub>, y<sub>1<\/sub>) is S<sub>1<\/sub> = 0.<\/p>\n<p>Theorem: The equation of the normal to the circle S = x<sup>2<\/sup> + y<sup>2<\/sup> + 2gx + 2fy + c = 0 at P(x<sub>1<\/sub>, y<sub>1<\/sub>) is<br \/>\n(y<sub>1<\/sub> + f) (x &#8211; x<sub>1<\/sub>) &#8211; (x<sub>1<\/sub> + g) (y &#8211; y<sub>1<\/sub>) = 0.<\/p>\n<p>Corollary: The equation of the normal to the circle x<sup>2<\/sup> + y<sup>2<\/sup> = a<sup>2<\/sup> at P(x<sub>1<\/sub>, y<sub>1<\/sub>) is y<sub>1<\/sub>x &#8211; x<sub>1<\/sub>y = 0.<\/p>\n<p>Theorem: The condition that the straight line lx + my + n = 0 may touch the circle x<sup>2<\/sup> + y<sup>2<\/sup> = a<sup>2<\/sup> is n<sup>2<\/sup> = a<sup>2<\/sup>(l<sup>2<\/sup> + m<sup>2<\/sup>) and the point of contact is \\(\\left(\\frac{-a^{2} 1}{n}, \\frac{-a^{2} m}{n}\\right)\\).<br \/>\nProof:<br \/>\nThe given line is lx + my + n = 0 &#8230;&#8230; (1)<br \/>\nThe given circle is x<sup>2<\/sup> + y<sup>2<\/sup> = r<sup>2<\/sup> &#8230;&#8230;. (2)<br \/>\nCentre C = (0, 0), radius = r<br \/>\nLine (1) is a tangent to the circle (2)<br \/>\n\u21d4 The perpendicular distance from the centre C to the line (1) is equal to the radius r.<br \/>\n\u21d4 \\(\\left|\\frac{0-n}{\\sqrt{1^{2}+m^{2}}}\\right|\\) = r<br \/>\n\u21d4 (n)<sup>2<\/sup> = r<sup>2<\/sup> (l<sup>2<\/sup> + m<sup>2<\/sup>)<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-20451\" src=\"https:\/\/apboardsolutions.com\/wp-content\/uploads\/2022\/08\/Inter-2nd-Year-Maths-2B-Circle-Formulas-4.png\" alt=\"Inter 2nd Year Maths 2B Circle Formulas 4\" width=\"109\" height=\"109\" \/><br \/>\nLet P(x<sub>1<\/sub>, y<sub>1<\/sub>) be the point of contact.<br \/>\nEquation of the tangent is S<sub>1<\/sub> = 0, \u21d2 x<sub>1<\/sub>x + y<sub>1<\/sub>y &#8211; r<sup>2<\/sup> = 0. &#8212;- (3)<br \/>\nEquations (1) and (3) are representing the same line, therefore, \\(\\frac{x_{1}}{l}=\\frac{y_{1}}{m}=\\frac{-a^{2}}{n}\\) \u21d2 x<sub>1<\/sub> = \\(\\frac{-a^{2} l}{n}\\), y<sub>1<\/sub> = \\(\\frac{-a^{2} m}{n}\\)<br \/>\nTherefore, point of contact is \\(\\left(\\frac{-a^{2} l}{n}, \\frac{-a^{2} m}{n}\\right)\\)<\/p>\n<p>Theorem: The condition for the straight line lx + my + n = 0 may be a tangent to the circle<br \/>\nx<sup>2<\/sup> + y<sup>2<\/sup> + 2gx + 2fy + c = 0 is (g<sup>2<\/sup> + f<sup>2<\/sup> &#8211; c) (l<sup>2<\/sup> + m<sup>2<\/sup>) = (lg + mf &#8211; n)<sup>2<\/sup>.<br \/>\nProof:<br \/>\nThe given line is lx + my + n = 0 &#8230;&#8230;.. (1)<br \/>\nThe given circle is x<sup>2<\/sup> + y<sup>2<\/sup> + 2gx + 2fy + c = 0 &#8230;&#8230;.. (2)<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-20451\" src=\"https:\/\/apboardsolutions.com\/wp-content\/uploads\/2022\/08\/Inter-2nd-Year-Maths-2B-Circle-Formulas-4.png\" alt=\"Inter 2nd Year Maths 2B Circle Formulas 4\" width=\"109\" height=\"109\" \/><br \/>\nCentre C = (- g, &#8211; f), radius r = \\(\\sqrt{\\mathrm{g}^{2}+\\mathrm{f}^{2}-\\mathrm{c}}\\)<br \/>\nLine (1) is a tangent to the circle (2)<br \/>\n\u21d4 The perpendicular distance from the centre C to the line (1) is equal to the radius r.<br \/>\n\u21d4 \\(\\left|\\frac{-\\lg -m f+c}{\\sqrt{1^{2}+m^{2}}}\\right|\\) = \\(\\sqrt{\\mathrm{g}^{2}+\\mathrm{f}^{2}-\\mathrm{c}}\\)<br \/>\n\u21d4 (lg + mf -n)<sup>2<\/sup> = (g<sup>2<\/sup> + f<sup>2<\/sup> &#8211; c) (l<sup>2<\/sup> + m<sup>2<\/sup>)<\/p>\n<p><img loading=\"lazy\" src=\"https:\/\/apboardsolutions.com\/wp-content\/uploads\/2022\/07\/AP-Board-Solutions.png\" alt=\"Inter 2nd Year Maths 2B Circle Formulas\" width=\"196\" height=\"17\" \/><\/p>\n<p>Corollary: The condition for the straight line y = mx + c to touch the circle<br \/>\nx<sup>2<\/sup> + y<sup>2<\/sup> = r<sup>2<\/sup> is c<sup>2<\/sup> = r<sup>2<\/sup>(1 + m<sup>2<\/sup>).<br \/>\nThe given line is y = mx + c i.e., mx &#8211; y + c = 0 &#8230; (1)<br \/>\nThe given circle is S = x<sup>2<\/sup> + y<sup>2<\/sup> = r<sup>2<\/sup><br \/>\nCentre C = (0,0), radius = r.<br \/>\nIf (1) is a tangent to the circle, then<br \/>\nRadius of the circle = perpendicular distance from centre of the circle to the line.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-20451\" src=\"https:\/\/apboardsolutions.com\/wp-content\/uploads\/2022\/08\/Inter-2nd-Year-Maths-2B-Circle-Formulas-4.png\" alt=\"Inter 2nd Year Maths 2B Circle Formulas 4\" width=\"109\" height=\"109\" \/><br \/>\n\u21d2 r = \\(\\frac{|c|}{\\sqrt{m^{2}+1}}\\) \u21d2 r<sup>2<\/sup> = \\(\\frac{c^{2}}{m^{2}+1}\\) \u21d2 r<sup>2<\/sup> (m<sup>2<\/sup> + 1) = c<sup>2<\/sup><\/p>\n<p>Corollary: If the straight line y = mx + c touches the circle x<sup>2<\/sup> + y<sup>2<\/sup> = r<sup>2<\/sup>, then their point of contact is \\(\\left(-\\frac{r^{2} m}{c}, \\frac{r^{2}}{c}\\right)\\).<br \/>\nProof:<br \/>\nThe given line is y = mx + c i.e., mx &#8211; y + c = 0 &#8230;&#8230;. (1)<br \/>\nThe given circle is S = x<sup>2<\/sup> + y<sup>2<\/sup> = r<sup>2<\/sup> &#8230;&#8230;. (2)<br \/>\nCentre C = (0, 0), radius = r<br \/>\nLet P(x<sub>1<\/sub>, y<sub>1<\/sub>) be the point of contact.<br \/>\nEquation of the tangent is S<sub>1<\/sub> = 0, \u21d2 x<sub>1<\/sub>x + y<sub>1<\/sub>y &#8211; r<sup>2<\/sup> = 0. &#8230;&#8230;&#8230;. (3)<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-20451\" src=\"https:\/\/apboardsolutions.com\/wp-content\/uploads\/2022\/08\/Inter-2nd-Year-Maths-2B-Circle-Formulas-4.png\" alt=\"Inter 2nd Year Maths 2B Circle Formulas 4\" width=\"109\" height=\"109\" \/><br \/>\nEquations (1) and (3) are representing the same line, therefore, \\(\\frac{x_{1}}{m}=\\frac{y_{1}}{-1}=\\frac{-r^{2}}{c}\\) \u21d2 x<sub>1<\/sub> = \\(\\frac{-r^{2} m}{c}\\), y<sub>1<\/sub> = \\(\\frac{r^{2}}{c}\\)<br \/>\nPoint of contact is (x<sub>1<\/sub>, y<sub>1<\/sub>) = \\(\\left(-\\frac{\\mathrm{r}^{2} \\mathrm{~m}}{\\mathrm{c}}, \\frac{\\mathrm{r}^{2}}{\\mathrm{c}}\\right)\\)<\/p>\n<p>Theorem: If P(x, y) is a point on the circle with centre C(\u03b1, \u03b2) and radius r, then x = \u03b1 + r cos \u03b8, y = \u03b2 + r sin \u03b8 where 0 \u2264 \u03b8 &lt; 2\u03c0.<\/p>\n<p>Note 1: The equations x = \u03b1 + r cos \u03b8, y = \u03b2 + r sin \u03b8, 0 \u2264 \u03b8 &lt; 2\u03c0 are called parametric equations of the circle with centre (\u03b1, \u03b2) and radius r.<\/p>\n<p>Note 2: A point on the circle x<sup>2<\/sup> + y<sup>2<\/sup> = r<sup>2<\/sup> is taken in the form (r cos\u03b8, r sin \u03b8). The point (r cos\u03b8, r sin \u03b8) is simply denoted as point \u03b8.<\/p>\n<p><img loading=\"lazy\" src=\"https:\/\/apboardsolutions.com\/wp-content\/uploads\/2022\/07\/AP-Board-Solutions.png\" alt=\"Inter 2nd Year Maths 2B Circle Formulas\" width=\"196\" height=\"17\" \/><\/p>\n<p>Theorem: The equation of the chord joining two points \u03b8<sub>1<\/sub> and \u03b8<sub>2<\/sub> on the circle<br \/>\nx<sup>2<\/sup> + y<sup>2<\/sup> + 2gx + 2fy + c = 0 is (x + g)cos\\(\\frac{\\theta_{1}+\\theta_{2}}{2}\\) + (y + f) sin \\(\\frac{\\theta_{1}+\\theta_{2}}{2}\\) = r cos \\(\\frac{\\theta_{1}+\\theta_{2}}{2}\\) where r is the radius of the circle.<\/p>\n<p>Note 1: The equation of the chord joining the points \u03b8<sub>1<\/sub> and \u03b8<sub>2<\/sub> on the circle x<sup>2<\/sup> + y<sup>2<\/sup> = r<sup>2<\/sup> is x cos\\(\\frac{\\theta_{1}+\\theta_{2}}{2}\\) + y sin\\(\\frac{\\theta_{1}+\\theta_{2}}{2}\\) = r cos\\(\\frac{\\theta_{1}-\\theta_{2}}{2}\\)<\/p>\n<p>Note 2: The equation of the tangent at P(\u03b8) on the circle (x + g) cos \u03b8 + (y + f) sin \u03b8 = \\(\\sqrt{g^{2}+f^{2}-c}\\).<\/p>\n<p>Note 3: The equation of the tangent at P(\u03b8) on the circle x<sup>2<\/sup> + y<sup>2<\/sup> = r<sup>2<\/sup> is x cos \u03b8 + y sin \u03b8 = r.<\/p>\n<p>Note 4: The equation of the normal at P(\u03b8) on the circle x<sup>2<\/sup> + y<sup>2<\/sup> = r<sup>2<\/sup> is x sin \u03b8 &#8211; y cos \u03b8 = r.<\/p>\n<p>Theorem:<br \/>\nIf a line passing through a point P(x<sub>1<\/sub>, y<sub>1<\/sub>) intersects the circle S = 0 at the points A and B then PA.PB = |S<sub>11<\/sub>|.<\/p>\n<p>Corollary:<br \/>\nIf the two lines a<sub>1<\/sub>x + b<sub>1<\/sub>y + c<sub>1<\/sub> = 0, a<sub>2<\/sub>x + b<sub>2<\/sub>y + c<sub>2<\/sub> = 0 meet the coordinate axes in four distinct points then those points are concyclic \u21d4 a<sub>1<\/sub>a<sub>2<\/sub> = b<sub>1<\/sub>b<sub>2<\/sub>.<\/p>\n<p>Corollary:<br \/>\nIf the lines a<sub>1<\/sub>x + b<sub>1<\/sub>y + c<sub>1<\/sub> = 0, a<sub>2<\/sub>x + b<sub>2<\/sub>y + c<sub>2<\/sub> = 0 meet the coordinate axes in four distinct concyclic points then the equation of the circle passing through these concyclic points is (a<sub>1<\/sub>x + b<sub>1<\/sub>y + c<sub>1<\/sub>) (a<sub>2<\/sub>x + b<sub>2<\/sub>y + c<sub>2<\/sub>) &#8211; (a<sub>1<\/sub>b<sub>2<\/sub> + a<sub>2<\/sub>b<sub>1<\/sub>)xy = 0.<\/p>\n<p><img loading=\"lazy\" src=\"https:\/\/apboardsolutions.com\/wp-content\/uploads\/2022\/07\/AP-Board-Solutions.png\" alt=\"Inter 2nd Year Maths 2B Circle Formulas\" width=\"196\" height=\"17\" \/><\/p>\n<p>Theorem:<br \/>\nTwo tangents can be drawn to a circle from an external point.<\/p>\n<p>Note:<br \/>\nIf m<sub>1<\/sub>, m<sub>1<\/sub> are the slopes of tangents drawn to the circle x<sup>2<\/sup> + y<sup>2<\/sup> = a<sup>2<\/sup> from an external point (x<sub>1<\/sub>, y<sub>1<\/sub>) then m<sub>1<\/sub> + m<sub>2<\/sub> = \\(\\frac{2 x_{1} y_{1}}{x_{1}^{2}-a^{2}}\\), m<sub>1<\/sub>m<sub>2<\/sub> = \\(\\frac{y_{1}^{2}-a^{2}}{x_{1}^{2}-a^{2}}\\).<\/p>\n<p>Theorem:<br \/>\nIf \u03b8 is the angle between the tangents through a point P to the circle S = 0 then tan = \\(\\frac{\\theta}{2}=\\frac{r}{\\sqrt{S_{11}}}\\) where r is the radius of the circle.<br \/>\nProof:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-20450\" src=\"https:\/\/apboardsolutions.com\/wp-content\/uploads\/2022\/08\/Inter-2nd-Year-Maths-2B-Circle-Formulas-5.png\" alt=\"Inter 2nd Year Maths 2B Circle Formulas 5\" width=\"397\" height=\"128\" \/><br \/>\nLet the two tangents from P to the circle S = 0 touch the circle at Q, R and \u03b8 be the angle between<br \/>\nthese two tangents. Let C be the centre of the circle. Now QC = r, PQ = \\(\\sqrt{S_{11}}\\) and \u2206PQC is a right angled triangle at Q.<br \/>\n\u2234 tan \\(\\frac{\\theta}{2}=\\frac{\\mathrm{QC}}{\\mathrm{PQ}}=\\frac{\\mathrm{r}}{\\sqrt{\\mathrm{S}_{11}}}\\)<\/p>\n<p>Theorem: The equation to the chord of contact of P(x<sub>1<\/sub>, y<sub>1<\/sub>) with respect to the circle S = 0 is S<sub>1<\/sub> = 0.<\/p>\n<p>Theorem: The equation of the polar of the point P(x<sub>1<\/sub>, y<sub>1<\/sub>) with respect to the circle S = 0 is S<sub>1<\/sub> = 0.<\/p>\n<p>Theorem: The pole of the line lx + my + n = 0 (n \u2260 0) with respect to x<sup>2<\/sup> + y<sup>2<\/sup> = a<sup>2<\/sup> is \\(\\left(-\\frac{1 a^{2}}{n},-\\frac{m a^{2}}{n}\\right)\\)<br \/>\nProof :<br \/>\nLet P(x<sub>1<\/sub>, y<sub>1<\/sub>) be the pole of lx + my + n = 0 &#8230;&#8230;. (1)<br \/>\nThe polar of P with respect to the circle is:<br \/>\nxx<sub>1<\/sub> + yy<sub>1<\/sub> &#8211; a<sup>2<\/sup> = 0<br \/>\nNow (1) and (2) represent the same line<br \/>\n\u2234 \\(\\frac{\\mathrm{x}_{1}}{\\ell}=\\frac{\\mathrm{y}_{1}}{\\mathrm{~m}}=\\frac{-\\mathrm{a}^{2}}{\\mathrm{n}}\\) \u21d2 x<sub>1<\/sub> = \\(\\frac{-\\mathrm{la}^{2}}{\\mathrm{n}}\\), y<sub>1<\/sub> = \\(\\frac{-\\mathrm{ma}^{2}}{\\mathrm{n}}\\)<br \/>\n\u2234 Pole P = \\(\\left(-\\frac{1 a^{2}}{n},-\\frac{m a^{2}}{n}\\right)\\)<\/p>\n<p><img loading=\"lazy\" src=\"https:\/\/apboardsolutions.com\/wp-content\/uploads\/2022\/07\/AP-Board-Solutions.png\" alt=\"Inter 2nd Year Maths 2B Circle Formulas\" width=\"196\" height=\"17\" \/><\/p>\n<p>Theorem: If the pole of the line lx + my + n = 0 with respect to the circle x<sup>2<\/sup> + y<sup>2<\/sup> + 2gx + 21y + c = 0 is (x<sub>1<\/sub>, y<sub>1<\/sub>) then \\(\\frac{x_{1}+g}{\\ell}=\\frac{y_{1}+f}{m}=\\frac{r^{2}}{\\lg +\\mathrm{mf}-\\mathrm{n}}\\) where r is the radius of the circle.<br \/>\nProof:<br \/>\nLet P(x<sub>1<\/sub>, y<sub>1<\/sub>) be the pole of the line lx + my + n = 0 &#8230;&#8230;. (1)<br \/>\nThe poiar of P with respect to S = 0 is S<sub>1<\/sub> = 0<br \/>\nxx<sub>1<\/sub> + yy<sub>1<\/sub> + g(x + x<sub>1<\/sub>) + f(y + y<sub>1<\/sub>) + c = 0<br \/>\n\u21d2 (x<sub>1<\/sub> + g)x + (y<sub>1<\/sub> + f) + gx<sub>1<\/sub> + fy<sub>1<\/sub> + c = 0 &#8230;&#8230;.. (2)<br \/>\nNow (1) and (2) represent the same line.<br \/>\n\u2234 \\(\\frac{\\mathrm{x}_{1}+\\mathrm{g}}{\\ell}=\\frac{\\mathrm{y}_{1}+\\mathrm{f}}{\\mathrm{m}}=\\frac{\\mathrm{gx} \\mathrm{x}_{1}+\\mathrm{gy} \\mathrm{y}_{1}+\\mathrm{c}}{\\mathrm{n}}\\) = k(say)<br \/>\n\\(\\frac{\\mathrm{x}_{1}+\\mathrm{g}}{\\ell}\\) = k \u21d2 x<sub>1<\/sub> + g = l k \u21d2 x<sub>1<\/sub> = lk &#8211; g<br \/>\n\\(\\frac{\\mathrm{y}_{1}+\\mathrm{f}}{\\mathrm{m}}\\) = k \u21d2 y<sub>1<\/sub> + f = m k \u21d2 y<sub>1<\/sub> = mk &#8211; f<br \/>\n\\(\\frac{g x_{1}+g y_{1}+c}{n}\\) = k \u21d2 gx<sub>1<\/sub> + gy<sub>1<\/sub> + c = nk<br \/>\n\u21d2 g(lk &#8211; g) + f(mk &#8211; f) + c = nk<br \/>\n\u21d2 k (lg + mf &#8211; n) = g<sup>2<\/sup> + f<sup>2<\/sup> &#8211; c = r<sup>2<\/sup> Where r is the radius of the circle \u21d2 k = \\(\\frac{r^{2}}{\\lg +\\mathrm{mf}-\\mathrm{n}}\\)<br \/>\n\u2234 \\(\\frac{\\mathrm{x}_{1}+\\mathrm{g}}{\\ell}=\\frac{\\mathrm{y}_{1}+\\mathrm{f}}{\\mathrm{m}}=\\frac{\\mathrm{r}^{2}}{\\lg +\\mathrm{mf}-\\mathrm{n}}\\)<\/p>\n<p>Theorem: The lines l<sub>1<\/sub>x + m<sub>1<\/sub>y + n<sub>1<\/sub> = 0 and l<sub>2<\/sub>x + m<sub>2<\/sub>y + n<sub>1<\/sub>y = 0 are conjugate with respect to the circle x<sup>2<\/sup> + y<sub>1<\/sub> + 2gx + 2fy + c = 0 iffr<sub>1<\/sub> (l<sub>1<\/sub>l<sub>2<\/sub> + m<sub>1<\/sub>m<sub>2<\/sub>) = (l<sub>1<\/sub>g + m<sub>1<\/sub>f &#8211; n<sub>1<\/sub>) (l<sub>2<\/sub>g + m<sub>2<\/sub>f &#8211; n<sub>2<\/sub>).<\/p>\n<p>Theorem: The condition for the lines l<sub>1<\/sub>x + m<sub>1<\/sub>y + n<sub>1<\/sub> = 0 and l<sub>2<\/sub>x + m<sub>2<\/sub>y + n<sub>2<\/sub> = 0 to be conjugate with respect to the circle x<sup>2<\/sup> + y<sup>2<\/sup> = a<sup>2<\/sup> is a<sup>2<\/sup>(l<sub>1<\/sub>l<sub>2<\/sub> + m<sub>1<\/sub>m<sub>2<\/sub>) = n<sub>1<\/sub>n<sub>2<\/sub>.<\/p>\n<p>Theorem: The equation of the chord of the circle S = 0 having P(x<sub>1<\/sub>, y<sub>1<\/sub>) as its midpoint is S<sub>1<\/sub> = S<sub>11<\/sub>.<\/p>\n<p>Theorem: The length of the chord of the circle S = 0 having P(x<sub>1<\/sub>, y<sub>1<\/sub>) as its midpoint is 2\\(\\sqrt{\\left|S_{11}\\right|}\\).<\/p>\n<p>Theorem: The equation to the pair of tangents to the circle<br \/>\nS = 0 from P(x<sub>1<\/sub>, y<sub>1<\/sub>) is S<sup>2<\/sup><sub>1<\/sub> = S<sub>11<\/sub>S.<br \/>\nProof:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-20449\" src=\"https:\/\/apboardsolutions.com\/wp-content\/uploads\/2022\/08\/Inter-2nd-Year-Maths-2B-Circle-Formulas-6.png\" alt=\"Inter 2nd Year Maths 2B Circle Formulas 6\" width=\"383\" height=\"202\" \/><br \/>\nLet the tangents from P to the circle S = 0 touch the circle at A and B.<br \/>\nEquation of AB is S<sub>1<\/sub> = 0.<br \/>\ni.e., x<sub>1<\/sub>x + y<sub>1<\/sub>y + g(x + x<sub>1<\/sub>) + f(y + y<sub>1<\/sub>) + c = 0 &#8212;&#8212;&#8212;- (i)<br \/>\nLet Q(x<sub>2<\/sub>, y<sub>2<\/sub>) be any point on these tangents. Now locus of Q will be the equation of the pair of tangents drawn from P.<br \/>\nThe line segment PQ is divided by the line AB in the ratio &#8211; S<sub>11<\/sub>:S<sub>22<\/sub><br \/>\n\u21d2 \\(\\frac{P B}{Q B}=\\left|\\frac{S_{11}}{S_{22}}\\right|\\) &#8212;&#8212;&#8212;&#8212; (ii)<br \/>\nBut PB = \\(\\sqrt{S_{11}}\\), QB = \\(\\sqrt{S_{22}}\\) \u21d2 \\(\\frac{P B}{Q B}=\\frac{\\sqrt{S_{11}}}{\\sqrt{S_{22}}}\\) &#8212;&#8212;&#8212;&#8212; (iii)<br \/>\nFrom (ii) and (iii) \u21d2 \\(\\frac{s_{11}^{2}}{s_{22}^{2}}=\\frac{S_{11}}{S_{22}}\\)<br \/>\n\u21d2 S<sub>11<\/sub>S<sub>22<\/sub> = S<sup>2<\/sup><sub>12<\/sub><br \/>\nHence locus of Q(x<sub>2<\/sub>, y<sub>2<\/sub>) is S<sub>11<\/sub>S = S<sup>2<\/sup><sub>12<\/sub><\/p>\n<p><img loading=\"lazy\" src=\"https:\/\/apboardsolutions.com\/wp-content\/uploads\/2022\/07\/AP-Board-Solutions.png\" alt=\"Inter 2nd Year Maths 2B Circle Formulas\" width=\"196\" height=\"17\" \/><\/p>\n<p>Touching Circles: Two circles S = 0 and S\u2019 = 0 are said to touch each other if they have a unique point P in common. The common point P is called point of contact of the circles S = 0 and S\u2019 = 0.<\/p>\n<p>Circle &#8211; Circle Properties: Let S = 0, S\u2019 = 0 be two circle with centres C<sub>1<\/sub>, C<sub>2<\/sub> and radii r<sub>1<\/sub>, r<sub>2<\/sub> respectively.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-20448\" src=\"https:\/\/apboardsolutions.com\/wp-content\/uploads\/2022\/08\/Inter-2nd-Year-Maths-2B-Circle-Formulas-7.png\" alt=\"Inter 2nd Year Maths 2B Circle Formulas 7\" width=\"531\" height=\"252\" \/><\/p>\n<ul>\n<li>If C<sub>1<\/sub>C<sub>2<\/sub> &gt; r<sub>1<\/sub> + r<sub>2<\/sub> then each circle lies completely outside the other circle.<\/li>\n<li>If C<sub>1<\/sub>C<sub>2<\/sub> = r<sub>1<\/sub> + r<sub>2<\/sub> then the two circles touch each other externally. The point of contact divides C<sub>1<\/sub>C<sub>2<\/sub> in the ratio r<sub>1<\/sub> : r<sub>2<\/sub> internally.<\/li>\n<li>If |r<sub>1<\/sub> &#8211; r<sub>2<\/sub>| &lt; C<sub>1<\/sub>C<sub>2<\/sub> &lt; r<sub>1<\/sub> + r<sub>2<\/sub> then the two circles intersect at two points P and Q. The chord \\(\\overline{\\mathrm{PQ}}\\) is called common chord of the circles.<\/li>\n<li>If C<sub>1<\/sub>C<sub>2<\/sub> = |r<sub>1<\/sub> &#8211; r<sub>2<\/sub>| then the two circles touch each other internally. The point of contact divides C<sub>1<\/sub>C<sub>2<\/sub> in the ratio r<sub>1<\/sub>: r<sub>2<\/sub> externally.<\/li>\n<li>If C<sub>1<\/sub>C<sub>1<\/sub> &lt; |r<sub>1<\/sub> &#8211; r<sub>2<\/sub>] then one circle lies completely inside the other circle.<\/li>\n<\/ul>\n<p>Common Tangents: A line L = 0 is said to be a common tangent to the circle S = 0, S\u2019 = 0 if L = 0 touches both the circles.<\/p>\n<p>Definition: A common tangent L = 0 of the circles S = 0, S\u2019= 0 is said to be a direct common tangent of the circles if the two circles S = 0, S\u2019 = 0 lie on the same side of L = 0.<\/p>\n<p>Centres of Similitude:<br \/>\nLet S = 0, S\u2019 = 0 be two circles.<\/p>\n<ul>\n<li>The point of intersection of direct common tangents of S = 0, S\u2019 = 0 is called external centre of similitude.<\/li>\n<li>The point of intersection of transverse common tangents of S = 0, S\u2019 = 0 is called internal centre of similitude.<\/li>\n<\/ul>\n<p>Theorem:<br \/>\nLet S = 0, S\u2019 = 0 be two circles with centres C<sub>1<\/sub>, C<sub>2<\/sub> and radii r<sub>1<\/sub>, r<sub>2<\/sub> respectively. If A<sub>1<\/sub> and A<sub>2<\/sub> are respectively the internal and external centres of similitude circles s = 0, S\u2019 = 0 then<\/p>\n<ul>\n<li>A<sub>1<\/sub> divides C<sub>1<\/sub>C<sub>2<\/sub> in the ratio r<sub>1<\/sub> : r<sub>2<\/sub> internally.<\/li>\n<li>A<sub>2<\/sub> divides C<sub>1<\/sub>C<sub>2<\/sub> in the ratio r<sub>1<\/sub> : r<sub>2<\/sub> internally.<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Use these Inter 2nd Year Maths 2B Formulas PDF Chapter 1 Circle to solve questions creatively. Intermediate 2nd Year Maths 2B Circle Formulas \u2192 The locus of a point in a plane such that its distance from a fixed point in the plane is always the same is called a circle. \u2192 The equation of &hellip;<\/p>\n","protected":false},"author":5,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"spay_email":""},"categories":[14],"tags":[],"jetpack_featured_media_url":"","_links":{"self":[{"href":"https:\/\/apboardsolutions.guru\/wp-json\/wp\/v2\/posts\/35220"}],"collection":[{"href":"https:\/\/apboardsolutions.guru\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/apboardsolutions.guru\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/apboardsolutions.guru\/wp-json\/wp\/v2\/users\/5"}],"replies":[{"embeddable":true,"href":"https:\/\/apboardsolutions.guru\/wp-json\/wp\/v2\/comments?post=35220"}],"version-history":[{"count":1,"href":"https:\/\/apboardsolutions.guru\/wp-json\/wp\/v2\/posts\/35220\/revisions"}],"predecessor-version":[{"id":35222,"href":"https:\/\/apboardsolutions.guru\/wp-json\/wp\/v2\/posts\/35220\/revisions\/35222"}],"wp:attachment":[{"href":"https:\/\/apboardsolutions.guru\/wp-json\/wp\/v2\/media?parent=35220"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/apboardsolutions.guru\/wp-json\/wp\/v2\/categories?post=35220"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/apboardsolutions.guru\/wp-json\/wp\/v2\/tags?post=35220"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}