SATHEE: PARABOLA-2 (Parabola)

PARABOLA-2 (Parabola)

A parabola is the locus of a point, whose distance from a fixed point is equal to the perpendicular distance from a fixed straight line.

Let $S$ be the focus, ${ZZ}^{'}$ be the directrix.

Consider $S (a, 0)$ and equation of ${ZZ}^{'}$ is $x + a = 0$ .

Axis of parabola is $x$ -axis.

Now according to definition.

$PS = PM$

$\begin{aligned} \sqrt{(x - a)^{2} + y^{2}} = | \frac{x + a}{\sqrt{1}} | \\ (x - a)^{2} + y^{2} = (x + a)^{2} \\ y^{2} = 4 a x \end{aligned}$

Vertex $(0, 0)$

Tangent of latus rectum $x = 0$

Extremities of latus rectum (a, 2a), (a, $- 2 a)$

Length of latus rectum. $= 4 a$

Focal distance (SP) $SP = PM = x + a$

Parametric form $x = a t^{2}, y = 2 a t, t$ is parameter.

Focal distance - the distance of a point on the parabola from the focus.

Focal chord - A chord of the parabola, which passes through the focus.

Double ordinate - A chord of the parabola perpendicular to the axis of the parabol(a).

Latus Rectum-A double ordinate passing through the focus or a focal chord perpendicular to the axis of parabol(a).

Perpendicular distance from focus on directrix = half the latus rectum.
Vertex is middle point of the focus and the point of intersection of directrix and axis.
Two parabolas are said to be equal if they have the same latus rectum.

Other Standard Forms of Parabola:

Equation of curve:	$y^{2} = - 4 a x$	$x^{2} = 4 ay$	$x^{2} = - 4 ay$
Vertex	$(0, 0)$	$(0, 0)$	$(0, 0)$
Focus	$(- a, 0)$	$(0, a)$	$(0, - a)$
Directrix	$x - a = 0$	$y + a = 0$	$y - a = 0$
Equation of axis	$y = 0$	$x = 0$	$x = 0$
Tangent of vertex	$x = 0$	$y = 0$	$y = 0$
Parametric form	$(- a t^{2}, 2 a t)$	$(2 a t, a t^{2})$	$(2 a t, - a t^{2})$

Position of a point with respet to Parabola

Equation of Parabola when vertex is shifte(d).

I. Axis is Parallel to $x$ -axis:

Let vertex $A$ be $(p, q)$ then equation of parabola be $(y - q)^{2} = 4 a (x - p)$ .

II. Axis is parallel to y-axis:

Let vertex $A$ be $(p, q)$ then equation of parabola is $(x - p)^{2} = 4 a (y - q)$

Example: 1 The equation of parabola is $y = a x^{2} + b x + c$ , find its vertex, focus, directrix,

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Solution:

Equation of a is, length of latus rectum.

$\begin{aligned} y = a x^{2} + \frac{b}{a} x + \frac{b^{2}}{4 a^{2}} - \frac{b^{2}}{4 a^{2}} + c \\ y - c = a x + {\frac{b}{2 a}}^{2} - \frac{b^{2}}{4 a^{2}} \end{aligned}$

$x + {\frac{b}{2 a}}^{2} = \frac{1}{a} y + \frac{b^{2}}{4 a} - c$

$X^{2} = 4 A Y$ where $X = x + \frac{b}{2 a}, Y = y + \frac{b^{2} - 4 a c}{4 a}, 4 A = \frac{1}{a}$

Vertex : $X = 0, Y = 0$ i.e. $- \frac{b}{2 a}, - \frac{b^{2} - 4 a c}{4 a}$

Focus: $X = 0, Y =$ A i.e. $- \frac{b}{2 a}, \frac{1 + 4 ac - b^{2}}{4 a}$

Equation of directrix: $y + \frac{b^{2} - 4 a c + 1}{4 a} = 0$

Equation of axis: $x + \frac{b}{2 a} = 0$

Length of latus rectum $= \frac{1}{a}$

Example: 2 The equation of parabola is $y^{2} = ax + ay$ . Find its vertex, focus, directrix, axis and length of latus rectum.

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Solution:

$y^{2} - a y = a x$

$y^{2} - a y + \frac{a^{2}}{4} = a x + \frac{a^{2}}{4}$

$y - {\frac{a}{2}}^{2} = a x + \frac{a}{4}$

$Y^{2} = 4 AX$

Where: $Y = y - \frac{a}{2}, X = x + \frac{a}{4}, 4 A = a$ ie. $A = \frac{a}{4}$

Vertex $- \frac{a}{4}, \frac{a}{2}$

Focus $0, \frac{a}{2}$

Directrix $x + \frac{a}{2} = 0$

Axis $y - \frac{a}{2} = 0$

Length of latus rectum $= a$

Practice questions

1. The equation of parabola whose focus is at $(- 1, - 2)$ and directrix is $x - 2 y + 3 = 0$ is

(a). $4 x^{2} - y^{2} - 4 x y + 4 x - 32 y - 16 = 0$

(b). $x^{2} + 4 y^{2} + 4 x y + x + 6 y + 16 = 0$

(c). $4 x^{2} + y^{2} + 4 x y + 4 x + 32 y + 16 = 0$

(d). $4 x^{2} + y^{2} - 4 x y + 4 x - 32 y - 1 = 0$

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Answer: (c)

2. The equation of parabola whose vertex is at $(4, - 1)$ and focus is $(4, - 3)$ is

(a). $y^{2} - 8 x + 8 y + 24 = 0$

(b). $x^{2} - 8 x + 8 y + 24 = 0$

(c). $y^{2} - 8 x - 8 y + 24 = 0$

(d). $x^{2} + 8 x - 8 y - 24 = 0$

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Answer: (b)

3. The focal distance of a point on the parabola $y^{2} = 8 x$ is 8 , then coordinates of the point (s) is/are

(a). $(4 \sqrt{3}, 6)$

(b). $(6, 4 \sqrt{3})$

(c). $(4 \sqrt{3}, - 6)$

(d). $(6, - 4 \sqrt{3})$

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Answer: (b, d)

4. The equation of the parabola whose focus is $(0, 0)$ and tangent at the vertex is $x - y + 1 = 0$ is

(a). $x^{2} + y^{2} + 2 x y - 4 x + 4 y - 4 = 0$

(b). $x^{2} + y^{2} + 4 x y + 4 x + 4 y + 4 = 0$

(c). $x^{2} + y^{2} - 4 x y + 4 x + 4 y - 4 = 0$

(d). $x^{2} + y^{2} - 4 x y - 4 x - 4 y - 4 = 0$

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Answer: (a)

Parabola

PARABOLA-2 (Parabola)

Other Standard Forms of Parabola:

Position of a point with respet to Parabola

Equation of Parabola when vertex is shifte(d).

Practice questions

Table of Contents

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Mentorship

NCERT Exercise Video Solution

Parabola

PARABOLA-2 (Parabola)

Other Standard Forms of Parabola:

Position of a point with respet to Parabola

Equation of Parabola when vertex is shifte(d).

Practice questions

Table of Contents

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Mentorship

NCERT Exercise Video Solution

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