{"id":35225,"date":"2022-08-25T11:31:34","date_gmt":"2022-08-25T06:01:34","guid":{"rendered":"https:\/\/apboardsolutions.guru\/?p=35225"},"modified":"2022-08-25T11:31:34","modified_gmt":"2022-08-25T06:01:34","slug":"inter-2nd-year-maths-2b-parabola-formulas","status":"publish","type":"post","link":"https:\/\/apboardsolutions.guru\/inter-2nd-year-maths-2b-parabola-formulas\/","title":{"rendered":"Inter 2nd Year Maths 2B Parabola Formulas"},"content":{"rendered":"

Use these Inter 2nd Year Maths 2B Formulas<\/a> PDF Chapter 3 Parabola to solve questions creatively.<\/p>\n

Intermediate 2nd Year Maths 2B Parabola Formulas<\/h2>\n

Definition:<\/strong>
\n\u2192 Conic section: If right circular solid cone is cut by a plane the section of it is called a Conic section.<\/p>\n

\u2192 Conic: The locus of a point whose distances from a fixed point and a fixed straight line are in constant ratio ‘e’ is called a conic.<\/p>\n

\u2192 Parabola : A conic with eccentricity 7 is called a parabola i.e., the locus of a point, whose distance from fixed point (focus) is equal to the distance from a fixed straight line (directrix) is called a parabola.<\/p>\n

\u2192 Axis of Parabola: The line passing through the vertex and the focus and perpendicular to directrix of the parabola is called the axis of the Parabola.<\/p>\n

\u2192 Equation of Parabola : General form – Let S(\u03b1, \u03b2) be the focus and ax + by + c = 0 be directrix then by definition the equation of parabola be,
\n\\(\\sqrt{(x-\\alpha)^{2}+(y-\\beta)^{2}}\\) = \\(\\left|\\frac{a x+b y+c}{\\sqrt{a^{2}+b^{2}}}\\right|\\)<\/p>\n

\"Inter<\/p>\n

Various forms of parabola:<\/strong>
\n\u2192 y2<\/sup> = 4ax
\nAxis : X – axis
\nFocus : (a, 0)
\nVertex : (0, 0)
\nEquation of directrix : x + a = 0
\n\"Inter<\/p>\n

\u2192 y2<\/sup>= – 4ax
\nAxis: x – axis
\nFocus: (- a, 0)
\nVertex : (0, 0)
\nEquation of directrix : x – a = 0
\n\"Inter<\/p>\n

\u2192 x2<\/sup> = 4ay (a > 0)
\nAxis: Y-axis Focus : (0, a)
\nVertex : (0, 0)
\nEquation of directrix: y + a = 0
\n\"Inter<\/p>\n

\"Inter<\/p>\n

\u2192 x2<\/sup> = -4ay(a>0)
\nAxis : Y axis
\nFocus : (0, – a)
\nVertex: (0, 0)
\nEquation of directrix : y – a = 0
\n\"Inter<\/p>\n

\u2192 (y – k)2<\/sup> = 4a (x – h) a > 0
\nAxis : y = k’ a line parallel to X- axis
\nFocus : (a + h, k)
\nVertex: (h, k)
\nEquation of directrix: x + a = h
\n\"Inter<\/p>\n

\u2192 (x – h)2<\/sup> = 4a (y – k) a > 0
\nAxis: x = h a line parallel to Y-axis
\nFocus: (h, a + k)
\nVertex: (h, k)
\nEquation of directrix: y + a = k
\n\"Inter<\/p>\n

\u2192 \\(\\sqrt{(x-h)^{2}+(y-k)^{2}}\\) = \\(\\left|\\frac{a x+b y+c}{\\sqrt{a^{2}+b^{2}}}\\right|\\)
\nAxis: b(x – h) – a(y – k) = 0
\nFocus: (h, k)
\nVertex: Some point A(fig)
\nEquation of directrix : ax + by + c = 0
\n\"Inter<\/p>\n

Chord: Line segment joining two points of a parabola is called a, chord of the parabola. If this chord passes through focus is called focal chord.
\nChord passing through focus and \u22a5 to axis is called latus rectum.
\nLength of latus rectum = 4a<\/p>\n

\"Inter<\/p>\n

Parametric form of parabola:
\nFor y2<\/sup> = 4ax
\nParametric form will be x = at2<\/sup>, y = 2at<\/p>\n

Definition:
\n\u2192 The locus of a point which moves in a plane so that its distance from a fixed point bears a constant ratio to its distance from a fixed straight line is called a conic section or conic. The fixed point is called focus, the fixed straight line is called directrix and the constant ratio \u2018e\u2019 is called eccentricity of the conic.<\/p>\n

If e = 1, then the conic is called a Parabola.
\nIf e < 1, then the conic is called an Ellipse. If e > 1, then the conic is called a hyperbola.
\nNote: The equation of a conic is of the form ax2<\/sup> + 2hxy + by2<\/sup> + 2gx + 2fy + c = 0.<\/p>\n

\u2192 Directrix of the Conic: A line L = 0 passing through the focus of a conic is said to be the principal axis of the conic if it is perpendicular to the directrix of the conic.<\/p>\n

\u2192 Vertices: The points of intersection of a conic and its principal axis are called vertices of the conic.<\/p>\n

\u2192 Centre: The midpoint o the line segment joining the vertices of a conic is called centre of the conic.<\/p>\n

\u2192 Note 1: If a conic has only one vertex then its centre coincides with the vertex.<\/p>\n

\u2192 Note 2: If a conic has two vertices then its centre does not coincide either of the vertices. In this case the conic is called a central conic.<\/p>\n

\u2192 Standard Form: A conic is said to be in the standard form if the principal axis of the conic is x-axis and the centre of the conic is the origin.<\/p>\n

Equation of a Parabola in Standard Form:
\nThe equation of a parabola in the standard form is y2<\/sup> = 4ax.
\nProof :
\nLet S be the focus and L = 0 be the directrix of the parabola.
\nLet P be a point on the parabola.
\nLet M, Z be the projections of P, S on the directrix L = 0 respectively.
\nLet N be the projection of P on SZ.
\nLet A be the midpoint of SZ.
\nTherefore, SA = AZ, \u21d2 A lies on the parabola. Let AS = a.
\nLet AS, the principal axis of the parabola as x-axis and Ay perpendicular to SZ as y-axis.
\nThen S = (a, 0) and the parabola is in the standard form.
\nLet P = (x1<\/sub>, y1<\/sub>).
\n\"Inter
\nNow PM = NZ = NA + AZ = x1<\/sub> + a
\nP lies on the parabola \u21d2 \\(\\frac{\\mathrm{PS}}{\\mathrm{PM}}\\) = 1 \u21d2 PS = PM
\n\u21d2 \\(\\sqrt{\\left(x_{1}-a\\right)^{2}+\\left(y_{1}-0\\right)^{2}}\\) = x1<\/sub> + a
\n\u21d2 (x1<\/sub> – a)2<\/sup> + y1<\/sub>2<\/sup> = (x1<\/sub> + a)2<\/sup>
\n\u21d2 y1<\/sub>2<\/sup> = (x1<\/sub> + a)2<\/sup> – (x1<\/sub> – a)2<\/sup>
\n\u21d2 y1<\/sub>2<\/sup> = 4ax1<\/sub>
\nThe locus of P is y2<\/sup> = 4ax.
\n\u2234 The equation to the parabola is y2<\/sup> = 4ax.<\/p>\n

\"Inter<\/p>\n

Nature of the Curve y2<\/sup> = 4ax.<\/strong><\/p>\n

(i) The curve is symmetric with respect to the x-axis.
\n\u2234 The principal axis (x-axis) is an axis of the parabola.<\/p>\n

(ii) y = 0 \u21d2 x = 0. Thus the curve meets x-axis at only one point (0, 0).
\nHence the parabola has only one vertex.<\/p>\n

(iii) If x < 0 then there exists no y \u2208 R. Thus the parabola does not lie in the second and third quadrants. (iv) If x > 0 then y2<\/sup> > 0 and hence y has two real values (positive and negative). Thus the parabola lies in the first and fourth quadrants.<\/p>\n

(v) x = 0 \u21d2 y2<\/sup> = 0 \u21d2 y = 0, 0. Thus y-axis meets the parabola in two coincident points and hence y-axis touches the parabola at (0, 0).<\/p>\n

(vi) As x \u2192 \u221e \u21d2 y2<\/sup> \u2192 \u221e \u21d2 y \u2192 \u00b1\u221e
\nThus the curve is not bounded (closed) on the right side of the y-axis.<\/p>\n

\u2192 Double Ordinate: A chord passing through a point P on the parabola and perpendicular to the principal axis of the parabola is called the double ordinate of the point P.<\/p>\n

\u2192 Focal Chord: A chord of the parabola passing through the focus is called a focal chord.<\/p>\n

\u2192 Latus Rectum: A focal chord of a parabola perpendicular to the principal axis of the parabola is called latus rectum. If the latus rectum meets the parabola in L and L\u2019, then LL\u2019 is called length of the latus rectum.<\/p>\n

Theorem: The length of the latus rectum of the parabola y2<\/sup> = 4ax is 4a.
\nProof:
\nLet LL\u2019 be the length of the latus rectum of the parabola y2<\/sup> = 4ax.
\n\"Inter
\nLet SL = 1, then L = (a, 1)
\nSince L is a point on the parabola y2<\/sup> = 4ax, therefore 12<\/sup> = 4a(a)
\n\u21d2 12<\/sup> = 4a2<\/sup> \u21d2 1 = 2a \u21d2 SL = 2a
\n\u2234 LL\u2019 = 2SL = 4a.<\/p>\n

\"Inter<\/p>\n

\u2192 Focal Distance: If P is a point on the parabola with focus S, then SP is called focal distance of P.<\/p>\n

Theorem: The focal distance of P(x1<\/sub>, y1<\/sub>) on the parabola y2<\/sup> = 4ax is x1<\/sub> + a.
\nNotation: We use the following notation in this chapter
\nS \u2261 y2<\/sup> – 4ax
\nS1<\/sub> \u2261 YY1<\/sub> – 2a(x + x1<\/sub>)
\nS11<\/sub> = S(x1<\/sub>, y1<\/sub>) \u2261 y1<\/sub>2<\/sup> – 4ax1<\/sub>
\nS12<\/sub> \u2261 y1<\/sub>y2<\/sub> – 2a(x1<\/sub> + x2<\/sub>)<\/p>\n

Note:
\nLet P(x1<\/sub>, y1<\/sub>) be a point and S \u2261 y2<\/sup> – 4ax = 0 be a parabola. Then<\/p>\n