Use these Inter 2nd Year Maths 2B Formulas PDF Chapter 1 Circle to solve questions creatively.

## Intermediate 2nd Year Maths 2B Circle Formulas

→ 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.

→ The equation of a circle with centre (h, k) and radius r is (x – h)2 + (y – k)2 = r2

→ The equation of a circle in standard form is x2 + y2 = r2.

→ The equation of a circle in general form is x2 + y2 + 2gx + 2fy + c = 0 and its centre is (-g, -f), radius is $$\sqrt{g^{2}+f^{2}-c}$$. → The intercept made by x2 + y2 + 2gx + 2fy + c = 0

• on X-axis is 2$$\sqrt{g^{2}-c}$$ if g2 > c.
• on Y-axis is 2$$\sqrt{f^{2}-c}$$ if f2 > c.

→ If the extremities of a diameter of a circle are (x1, y1) and (x2, y2) then its equation is (x – x1) (x – x2) + (y – y1) (y – y2) = 0

→ The equation of a circle passing through three non-collinear points (x1, y1), (x2, y2) and (x3, y3) is
$$\left|\begin{array}{lll} x_{1} & y_{1} & 1 \\ x_{2} & y_{2} & 1 \\ x_{3} & y_{3} & 1 \end{array}\right|$$ = (x2 + y2) + $$\left|\begin{array}{lll} c_{1} & y_{1} & 1 \\ c_{2} & y_{2} & 1 \\ c_{3} & y_{3} & 1 \end{array}\right|$$ x + $$\left|\begin{array}{lll} x_{1} & C_{1} & 1 \\ x_{2} & C_{2} & 1 \\ x_{3} & C_{3} & 1 \end{array}\right|$$ y + $$\left|\begin{array}{lll} x_{1} & y_{1} & C_{1} \\ x_{2} & y_{2} & C_{2} \\ x_{3} & y_{3} & C_{3} \end{array}\right|$$ = 0.
where ci = – (xi2 + yi2)

→ The centre of the circle passing through three non-collinear points (x1, y1), (x2, y2) and (x3, y3) is
$$\left[\frac{\left|\begin{array}{lll} c_{1} & y_{1} & 1 \\ c_{2} & y_{2} & 1 \\ c_{3} & y_{3} & 1 \end{array}\right|}{(-2)\left|\begin{array}{lll} x_{1} & y_{1} & 1 \\ x_{2} & y_{2} & 1 \\ x_{3} & y_{3} & 1 \end{array}\right|}, \frac{\left|\begin{array}{lll} x_{1} & c_{1} & 1 \\ x_{2} & c_{2} & 1 \\ x_{3} & c_{3} & 1 \end{array}\right|}{(-2)\left|\begin{array}{lll} x_{1} & y_{1} & 1 \\ x_{2} & y_{2} & 1 \\ x_{3} & y_{3} & 1 \end{array}\right|}\right]$$ → The parametric equations of a circle with centre (h, k) and radius (r ≥ 0) are given by
x = h + r cos θ
y = k + r sin θ 0 ≤ 6 < 2π.

→ A point P(x1, y1) is an interior point or on the circumference or an exterior point of a circles S = 0 ⇔ S11 $$\frac{<}{>}$$ 0.

→ The power of P(x1, y1) with respect to the circle S = 0 is S11.

→ A point P(x1, y1) is an interior point or on the circumference or exterior point of the circle S = 0 ⇔ the power of P with respect to S = 0 is negative, zero and positive.

→ If a straight line through a point P(x1, y1) meets the circle S = 0 at A and B then the power of P is equal to PA. PB.

→ The length of the tangent from P(x1, y1) to S = 0 is $$\sqrt{S_{11}}$$.

→ The straight line l = 0 intersects, touches or does not meet the circles = 0 according as l < r, l = r or l > r where l is the perpendicular distance from the centre of the circle to the line l = 0 and r is the radius.

→ For every real value of m the straight line y = mx ± r $$\sqrt{1+m^{2}}$$ is a tangent to the circle x2 + y2 = r2.

→ If r is the radius of the circle S = x2 + y2 + 2gx + 2fy + c = 0 then for every real value of m the straight line y + f = m(n + g) ±r + m2 will be a tangent to the circle.

→ If P(x1, y1) and Q(x2, y2) are two points on the circle S = 0 then the secant’s $$(\stackrel{\leftrightarrow}{P Q})$$ equation is S1 + S2 = S12

→ The equation of tangent at (x1, y1) of the circle S = 0 is S1 = 0.

→ If θ1, θ2 are two points on S = x2 + y2 + 2gx + 2fy + c = 0 then the equation of the chord joining the points θ1, θ2 is
(x + g) cos $$\left(\frac{\theta_{1}+\theta_{2}}{2}\right)$$ + (y + f) sin $$\left(\frac{\theta_{1}+\theta_{2}}{2}\right)$$ = r cos $$\left(\frac{\theta_{1}-\theta_{2}}{2}\right)$$

→ The equation of the tangent at θ of the circle S = 0 is (x + g) cos θ + (y + f) sin θ = r.

→ The equation of normal at (x1, y1) of the circle
S = 0 is (x – x1) (y1 + f) – (y – y1) (x1 + g) = 0. → The chord of contact of P(x1 y1) (exterior point) with respect to S = 0 is S1 = 0.

→ The equation of the polar of a point P(x1, y1) with respect to S = 0 is S1 = 0.

 P(x1, y1) Tangent at P Chord of contact at p Polar of P (i) Interior of the circle Does not exist Does not exist (not defined) S1 = 0 (P is different from the centre of the circle) (ii) On the circle S1 = 0 S1 = 0 S1 = 0 (iii) Exterior of the circle Does not exist S1 = 0 S1 = 0

→ The pole of lx + my + n = 0 with respect to S = 0 is
$$\left(-g+\frac{l r^{2}}{l g+m f-n},-f+\frac{m r^{2}}{l g+m f-n}\right)$$

→ Where r is the radius of the circle. The polar of P(x1, y1) with respect to S = 0 passes through Q(x2, y2) ⇔ the polar of Q with respect to S – 0 passes through P.

→ The points (x1, y1) and (x2, y2) are conjugate points with respect to S = 0 if S12 = 0

→ Two lines l1x = m1y + n1 = 0, l2x + m2y + n2 = 0 are conjugate with respect to x2 + y2 = a2 ⇔ (l1l2 + m1m2) = n1n2

→ Two points P, Q are said to be inverse points with respect to S = 0 if CP. CQ = r2 where C is the centre and r is the radius of the circle S = 0.

→ If (x1, y1) is the mid-point of a chord of the circle S = 0 then its chord equation is S1 = S11.

→ The pair of common tangents to the circles S = 0, S’ = 0 touching at a point on the lines segment $$\overline{\mathrm{C}_{1} \mathrm{C}_{2}}$$ (C1, C2 are centres of the circles) is called transverse pair of common tangents.

→ The pair of common tangents to the circles S = 0, S’ = 0 intersecting at a point not in $$\overline{\mathrm{C}_{1} \mathrm{C}_{2}}$$ is called as direct pair of common tangents. → The point of intersection of transvese (direct) common tangents is called internal (external) Centre of similitude.

 Situation No of common tangents 1. $$\overline{C_{1} C_{2}}$$ > r1 + r2 4 2. r1 + r2 = $$\overline{C_{1} C_{2}}$$ 3 3. |r1 – r2| < $$\overline{C_{1} C_{2}}$$ < r1 + r2 2 4. C1C2 = |r1 – r2| 1 5. C1C2 < |r1 – r2| 0

→ The combined equation of the pair of tangents drawn from an external point P(x1, y1) to the circle S = 0 is SS11 = S21.

Equation of a Circle:
The equation of the circle with centre C (h, k) and radius r is (x – h)2 + (y – k)2 = r2.
Proof:
Let P(x1, y1) be a point on the circle.
P lies in the circle ⇔ PC = r ⇔ $$\sqrt{\left(\mathrm{x}_{1}-\mathrm{h}\right)^{2}+\left(\mathrm{y}_{1}-\mathrm{k}\right)^{2}}$$ = r
⇔ (x1 – h)2 + (y1 – k)2 = r2. The locus of P is (x – h)2 + (y – k)2 = r2.
∴ The equation of the circle is (x – h)2 + (y – k)2 = r2.

Note: The equation of a circle with centre origin and radius r is (x – 0)2 + (y – 0)2 = r2
i.e., x2 + y2 = r2 which is the standard equation of the circle.

Note: On expanding equation (1), the equation of a circle is of the form x2 + y2 + 2gx + 2fy + c = 0.

Theorem: If g2 + f2 – c ≥ 0, then the equation x2 + y2 + 2gx + 2fy + c = 0 represents a circle with centre (- g, – f) and radius $$\sqrt{g^{2}+f^{2}-c}$$.
Note: If ax2 + ay2 + 2gx + 2fy + c = 0 represents a circle, then its centre = $$\left(-\frac{g}{a},-\frac{f}{a}\right)$$ and its radius $$\frac{\sqrt{\mathrm{g}^{2}+\mathrm{f}^{2}-\mathrm{ac}}}{|\mathrm{a}|}$$.

Theorem: The equation of a circle having the line segment joining A(x1, y1) and B(x2, y2) as diameter is (x – x1) (x – x2) + (y – y1) (y – y2) = 0. Let P(x, y) be any point on the circle. Given points A(x1, y1) and B(x2, y2).
Now ∠APB = $$\frac{\pi}{2}$$. (Angle in a semi circle.)
Slope of AP. Slope of BP = – 1
⇒ $$\frac{y-y_{1}}{x-x_{1}} \frac{y-y_{2}}{x-x_{2}}$$ = – 1
⇒ (y – y2) (y – y1) = – (x – x2) (x – x1) = 0
⇒ (x – x2) (x – x1) + (y – y2) (y – y1) = 0

Definition: Two circles are said to be concentric if they have same center. The equation of the circle concentric with the circle x2 + y2 + 2gx + 2fy + c = 0 is of the form x2 + y2 + 2gx + 2fy + k = 0.
The equation of the concentric circles differs by constant only.

Parametric Equations of A Circle:

Theorem: If P(x, y) is a point on the circle with centre C(α, β) and radius r, then x = α + r cosθ, y = β + r sin θ where 0 ≤ θ < 2π.

Note: The equations x = α + r cos θ, y = + r sin θ, 0 ≤ θ < 2π are called parametric equations of the circle with centre (α, β) and radius r.

Note: A point on the circle x2 + y2 = r2 is taken in the form (r cos θ, r sin θ). The point (r cos θ, r sin θ) is simply denoted as point θ.

Theorem:
(1) If g2 – c > 0 then the intercept made on the x axis by the circle x2 + y2 + 2gx + 2fy + c = 0 is 2$$\sqrt{g^{2}-a c}$$
(2) If f2 – c >0 then the intercept made on the y axis by the circle x2 + y2 + 2gx + 2fy + c = 0 is 2$$\sqrt{f^{2}-b c}$$

Note: The condition for the x-axis to touch the circle
x2 + y2 + 2gx + 2fy + c = 0 (c > 0) is g2 = c.

Note: The condition of the y-axis to touch the circle
x2 + y2 + 2gx + 2fy + c = 0 (c > 0) is f2 = c.

Position of Point:
Let S = 0 be a circle and P(x1, y1) be a point I in the plane of the circle. Then

• P lies inside the circle S = 0 ⇔ S11 < 0
• P lies in the circle S = 0 ⇔ S11 = 0
• Plies outside the circle S = 0 ⇔ S11 = 0

Power of a Point:
Let S = 0 be a circle with centre C and radius r. Let P be a point. Then CP2 – r2 is called power of P with respect to the circle S = 0.

Theorem: The power of a point P(x1, y1) with respect to the circle S = 0 is S11.

Theorem: The length of the tangent drawn from an external point P(x1, y1) to the circle s = 0 is $$\sqrt{\mathrm{S}_{11}}$$. Theorem: The equation of the tangent to the circle S = 0 at P(x1, y1) is S1 = 0.

Theorem: The equation of the normal to the circle S = x2 + y2 + 2gx + 2fy + c = 0 at P(x1, y1) is
(y1 + f) (x – x1) – (x1 + g) (y – y1) = 0.

Corollary: The equation of the normal to the circle x2 + y2 = a2 at P(x1, y1) is y1x – x1y = 0.

Theorem: The condition that the straight line lx + my + n = 0 may touch the circle x2 + y2 = a2 is n2 = a2(l2 + m2) and the point of contact is $$\left(\frac{-a^{2} 1}{n}, \frac{-a^{2} m}{n}\right)$$.
Proof:
The given line is lx + my + n = 0 …… (1)
The given circle is x2 + y2 = r2 ……. (2)
Centre C = (0, 0), radius = r
Line (1) is a tangent to the circle (2)
⇔ The perpendicular distance from the centre C to the line (1) is equal to the radius r.
⇔ $$\left|\frac{0-n}{\sqrt{1^{2}+m^{2}}}\right|$$ = r
⇔ (n)2 = r2 (l2 + m2) Let P(x1, y1) be the point of contact.
Equation of the tangent is S1 = 0, ⇒ x1x + y1y – r2 = 0. —- (3)
Equations (1) and (3) are representing the same line, therefore, $$\frac{x_{1}}{l}=\frac{y_{1}}{m}=\frac{-a^{2}}{n}$$ ⇒ x1 = $$\frac{-a^{2} l}{n}$$, y1 = $$\frac{-a^{2} m}{n}$$
Therefore, point of contact is $$\left(\frac{-a^{2} l}{n}, \frac{-a^{2} m}{n}\right)$$

Theorem: The condition for the straight line lx + my + n = 0 may be a tangent to the circle
x2 + y2 + 2gx + 2fy + c = 0 is (g2 + f2 – c) (l2 + m2) = (lg + mf – n)2.
Proof:
The given line is lx + my + n = 0 …….. (1)
The given circle is x2 + y2 + 2gx + 2fy + c = 0 …….. (2) Centre C = (- g, – f), radius r = $$\sqrt{\mathrm{g}^{2}+\mathrm{f}^{2}-\mathrm{c}}$$
Line (1) is a tangent to the circle (2)
⇔ The perpendicular distance from the centre C to the line (1) is equal to the radius r.
⇔ $$\left|\frac{-\lg -m f+c}{\sqrt{1^{2}+m^{2}}}\right|$$ = $$\sqrt{\mathrm{g}^{2}+\mathrm{f}^{2}-\mathrm{c}}$$
⇔ (lg + mf -n)2 = (g2 + f2 – c) (l2 + m2) Corollary: The condition for the straight line y = mx + c to touch the circle
x2 + y2 = r2 is c2 = r2(1 + m2).
The given line is y = mx + c i.e., mx – y + c = 0 … (1)
The given circle is S = x2 + y2 = r2
Centre C = (0,0), radius = r.
If (1) is a tangent to the circle, then
Radius of the circle = perpendicular distance from centre of the circle to the line. ⇒ r = $$\frac{|c|}{\sqrt{m^{2}+1}}$$ ⇒ r2 = $$\frac{c^{2}}{m^{2}+1}$$ ⇒ r2 (m2 + 1) = c2

Corollary: If the straight line y = mx + c touches the circle x2 + y2 = r2, then their point of contact is $$\left(-\frac{r^{2} m}{c}, \frac{r^{2}}{c}\right)$$.
Proof:
The given line is y = mx + c i.e., mx – y + c = 0 ……. (1)
The given circle is S = x2 + y2 = r2 ……. (2)
Centre C = (0, 0), radius = r
Let P(x1, y1) be the point of contact.
Equation of the tangent is S1 = 0, ⇒ x1x + y1y – r2 = 0. ………. (3) Equations (1) and (3) are representing the same line, therefore, $$\frac{x_{1}}{m}=\frac{y_{1}}{-1}=\frac{-r^{2}}{c}$$ ⇒ x1 = $$\frac{-r^{2} m}{c}$$, y1 = $$\frac{r^{2}}{c}$$
Point of contact is (x1, y1) = $$\left(-\frac{\mathrm{r}^{2} \mathrm{~m}}{\mathrm{c}}, \frac{\mathrm{r}^{2}}{\mathrm{c}}\right)$$

Theorem: If P(x, y) is a point on the circle with centre C(α, β) and radius r, then x = α + r cos θ, y = β + r sin θ where 0 ≤ θ < 2π.

Note 1: The equations x = α + r cos θ, y = β + r sin θ, 0 ≤ θ < 2π are called parametric equations of the circle with centre (α, β) and radius r.

Note 2: A point on the circle x2 + y2 = r2 is taken in the form (r cosθ, r sin θ). The point (r cosθ, r sin θ) is simply denoted as point θ. Theorem: The equation of the chord joining two points θ1 and θ2 on the circle
x2 + y2 + 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.

Note 1: The equation of the chord joining the points θ1 and θ2 on the circle x2 + y2 = r2 is x cos$$\frac{\theta_{1}+\theta_{2}}{2}$$ + y sin$$\frac{\theta_{1}+\theta_{2}}{2}$$ = r cos$$\frac{\theta_{1}-\theta_{2}}{2}$$

Note 2: The equation of the tangent at P(θ) on the circle (x + g) cos θ + (y + f) sin θ = $$\sqrt{g^{2}+f^{2}-c}$$.

Note 3: The equation of the tangent at P(θ) on the circle x2 + y2 = r2 is x cos θ + y sin θ = r.

Note 4: The equation of the normal at P(θ) on the circle x2 + y2 = r2 is x sin θ – y cos θ = r.

Theorem:
If a line passing through a point P(x1, y1) intersects the circle S = 0 at the points A and B then PA.PB = |S11|.

Corollary:
If the two lines a1x + b1y + c1 = 0, a2x + b2y + c2 = 0 meet the coordinate axes in four distinct points then those points are concyclic ⇔ a1a2 = b1b2.

Corollary:
If the lines a1x + b1y + c1 = 0, a2x + b2y + c2 = 0 meet the coordinate axes in four distinct concyclic points then the equation of the circle passing through these concyclic points is (a1x + b1y + c1) (a2x + b2y + c2) – (a1b2 + a2b1)xy = 0. Theorem:
Two tangents can be drawn to a circle from an external point.

Note:
If m1, m1 are the slopes of tangents drawn to the circle x2 + y2 = a2 from an external point (x1, y1) then m1 + m2 = $$\frac{2 x_{1} y_{1}}{x_{1}^{2}-a^{2}}$$, m1m2 = $$\frac{y_{1}^{2}-a^{2}}{x_{1}^{2}-a^{2}}$$.

Theorem:
If θ 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.
Proof: Let the two tangents from P to the circle S = 0 touch the circle at Q, R and θ be the angle between
these two tangents. Let C be the centre of the circle. Now QC = r, PQ = $$\sqrt{S_{11}}$$ and ∆PQC is a right angled triangle at Q.
∴ tan $$\frac{\theta}{2}=\frac{\mathrm{QC}}{\mathrm{PQ}}=\frac{\mathrm{r}}{\sqrt{\mathrm{S}_{11}}}$$

Theorem: The equation to the chord of contact of P(x1, y1) with respect to the circle S = 0 is S1 = 0.

Theorem: The equation of the polar of the point P(x1, y1) with respect to the circle S = 0 is S1 = 0.

Theorem: The pole of the line lx + my + n = 0 (n ≠ 0) with respect to x2 + y2 = a2 is $$\left(-\frac{1 a^{2}}{n},-\frac{m a^{2}}{n}\right)$$
Proof :
Let P(x1, y1) be the pole of lx + my + n = 0 ……. (1)
The polar of P with respect to the circle is:
xx1 + yy1 – a2 = 0
Now (1) and (2) represent the same line
∴ $$\frac{\mathrm{x}_{1}}{\ell}=\frac{\mathrm{y}_{1}}{\mathrm{~m}}=\frac{-\mathrm{a}^{2}}{\mathrm{n}}$$ ⇒ x1 = $$\frac{-\mathrm{la}^{2}}{\mathrm{n}}$$, y1 = $$\frac{-\mathrm{ma}^{2}}{\mathrm{n}}$$
∴ Pole P = $$\left(-\frac{1 a^{2}}{n},-\frac{m a^{2}}{n}\right)$$ Theorem: If the pole of the line lx + my + n = 0 with respect to the circle x2 + y2 + 2gx + 21y + c = 0 is (x1, y1) 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.
Proof:
Let P(x1, y1) be the pole of the line lx + my + n = 0 ……. (1)
The poiar of P with respect to S = 0 is S1 = 0
xx1 + yy1 + g(x + x1) + f(y + y1) + c = 0
⇒ (x1 + g)x + (y1 + f) + gx1 + fy1 + c = 0 …….. (2)
Now (1) and (2) represent the same line.
∴ $$\frac{\mathrm{x}_{1}+\mathrm{g}}{\ell}=\frac{\mathrm{y}_{1}+\mathrm{f}}{\mathrm{m}}=\frac{\mathrm{gx} \mathrm{x}_{1}+\mathrm{gy} \mathrm{y}_{1}+\mathrm{c}}{\mathrm{n}}$$ = k(say)
$$\frac{\mathrm{x}_{1}+\mathrm{g}}{\ell}$$ = k ⇒ x1 + g = l k ⇒ x1 = lk – g
$$\frac{\mathrm{y}_{1}+\mathrm{f}}{\mathrm{m}}$$ = k ⇒ y1 + f = m k ⇒ y1 = mk – f
$$\frac{g x_{1}+g y_{1}+c}{n}$$ = k ⇒ gx1 + gy1 + c = nk
⇒ g(lk – g) + f(mk – f) + c = nk
⇒ k (lg + mf – n) = g2 + f2 – c = r2 Where r is the radius of the circle ⇒ k = $$\frac{r^{2}}{\lg +\mathrm{mf}-\mathrm{n}}$$
∴ $$\frac{\mathrm{x}_{1}+\mathrm{g}}{\ell}=\frac{\mathrm{y}_{1}+\mathrm{f}}{\mathrm{m}}=\frac{\mathrm{r}^{2}}{\lg +\mathrm{mf}-\mathrm{n}}$$

Theorem: The lines l1x + m1y + n1 = 0 and l2x + m2y + n1y = 0 are conjugate with respect to the circle x2 + y1 + 2gx + 2fy + c = 0 iffr1 (l1l2 + m1m2) = (l1g + m1f – n1) (l2g + m2f – n2).

Theorem: The condition for the lines l1x + m1y + n1 = 0 and l2x + m2y + n2 = 0 to be conjugate with respect to the circle x2 + y2 = a2 is a2(l1l2 + m1m2) = n1n2.

Theorem: The equation of the chord of the circle S = 0 having P(x1, y1) as its midpoint is S1 = S11.

Theorem: The length of the chord of the circle S = 0 having P(x1, y1) as its midpoint is 2$$\sqrt{\left|S_{11}\right|}$$.

Theorem: The equation to the pair of tangents to the circle
S = 0 from P(x1, y1) is S21 = S11S.
Proof: Let the tangents from P to the circle S = 0 touch the circle at A and B.
Equation of AB is S1 = 0.
i.e., x1x + y1y + g(x + x1) + f(y + y1) + c = 0 ———- (i)
Let Q(x2, y2) be any point on these tangents. Now locus of Q will be the equation of the pair of tangents drawn from P.
The line segment PQ is divided by the line AB in the ratio – S11:S22
⇒ $$\frac{P B}{Q B}=\left|\frac{S_{11}}{S_{22}}\right|$$ ———— (ii)
But PB = $$\sqrt{S_{11}}$$, QB = $$\sqrt{S_{22}}$$ ⇒ $$\frac{P B}{Q B}=\frac{\sqrt{S_{11}}}{\sqrt{S_{22}}}$$ ———— (iii)
From (ii) and (iii) ⇒ $$\frac{s_{11}^{2}}{s_{22}^{2}}=\frac{S_{11}}{S_{22}}$$
⇒ S11S22 = S212
Hence locus of Q(x2, y2) is S11S = S212 Touching Circles: Two circles S = 0 and S’ = 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’ = 0.

Circle – Circle Properties: Let S = 0, S’ = 0 be two circle with centres C1, C2 and radii r1, r2 respectively. • If C1C2 > r1 + r2 then each circle lies completely outside the other circle.
• If C1C2 = r1 + r2 then the two circles touch each other externally. The point of contact divides C1C2 in the ratio r1 : r2 internally.
• If |r1 – r2| < C1C2 < r1 + r2 then the two circles intersect at two points P and Q. The chord $$\overline{\mathrm{PQ}}$$ is called common chord of the circles.
• If C1C2 = |r1 – r2| then the two circles touch each other internally. The point of contact divides C1C2 in the ratio r1: r2 externally.
• If C1C1 < |r1 – r2] then one circle lies completely inside the other circle.

Common Tangents: A line L = 0 is said to be a common tangent to the circle S = 0, S’ = 0 if L = 0 touches both the circles.

Definition: A common tangent L = 0 of the circles S = 0, S’= 0 is said to be a direct common tangent of the circles if the two circles S = 0, S’ = 0 lie on the same side of L = 0.

Centres of Similitude:
Let S = 0, S’ = 0 be two circles.

• The point of intersection of direct common tangents of S = 0, S’ = 0 is called external centre of similitude.
• The point of intersection of transverse common tangents of S = 0, S’ = 0 is called internal centre of similitude.

Theorem:
Let S = 0, S’ = 0 be two circles with centres C1, C2 and radii r1, r2 respectively. If A1 and A2 are respectively the internal and external centres of similitude circles s = 0, S’ = 0 then

• A1 divides C1C2 in the ratio r1 : r2 internally.
• A2 divides C1C2 in the ratio r1 : r2 internally.