Name: Conic Sections: The Parabola
Uploaded: 2019-07-19
Duration: 9 min 43 s
Description: This video defines a parabola and explains how to graph a parabola in standard form, which will help you to solve questions in exam. Watch and learn.

Question

A variable chord

P Q

of a parabola

y^{2} = 4 a x

, subtends a right angle at the vertex. Show that it always passes through a fixed point. Also show that the locus of the point of intersection of the tangents at

P

and

Q

is a straight line. Find the locus of the mid point of

P Q

.

Accepted Answer

Given parabola is

$y^{2} = 4 a x$ where vertex is $O (0, 0)$ .

Let $P$ be $(a t_{1}^{2}, 2 a t) & Q$ be $(a t_{2}^{2}, 2 a t_{2})$

$∵ P Q$ makes right angle at the vertex $O (0, 0)$ .

Slope of $O P = \frac{2 a t_{1} - 0}{a t_{1}^{2} - 0} = \frac{2}{t_{1}}$

Slope of $O Q = \frac{2 a t_{2} - 0}{t_{2}^{2} - 0} = \frac{2}{t_{2}}$

$∵ Slope of O P \times Slope of O Q = - 1$

$\Rightarrow \frac{2}{t_{1}} \cdot \frac{2}{t_{2}} = - 1$

$\Rightarrow t_{1} t_{2} = - 4 . . . . (1)$

Now equation of chord $P Q$ is

$y - 2 a t_{1} = \frac{2 a t_{2} - 2 a t_{1}}{a t_{2}^{2} - a t_{1}^{2}} (x - a t_{1}^{2})$

$\Rightarrow y - 2 a t_{1} = \frac{2}{t_{1} + t_{2}} (x - a t_{1}^{2})$

At the axis of the parabola put $y = 0$

$0 - 2 a t_{1} = \frac{2}{t_{1} + t_{2}} (x - a t_{1}^{2})$

$\Rightarrow - a t_{1}^{2} - a t_{1} t_{2} = x - a t_{1}^{2}$

$\Rightarrow x = 4 a$

$∴$ Chord $P Q$ passes through $(4 a, 0)$

As, we know that point of intersection of tangents at $P (a t_{1}^{2}, 2 a t_{1}) & Q (a t_{2}^{2}, 2 a t_{2})$ is given by $(a t_{1} t_{2}, a (t_{1} + t_{2}))$ .

But $t_{1} \cdot t_{2} = - 4$ from $(1)$

$∴$ Point of intersection of tangents at $P & Q$ is $(- 4 a, a (t_{1} + t_{2}))$ Which lies on $x = - 4 a$ for all values of $t_{1} & t_{2}$

Hence, locus of point of intersection of tangents at $P & Q$ is a straight line i.e. $x + 4 a = 0$ .

Now, let mid-point of $P Q$ be $M (h, k)$

$∴ h = \frac{a t_{1}^{2} + a t_{2}^{2}}{2} & k = \frac{2 a t_{1} + 2 a t_{2}}{2}$

$\Rightarrow 2 h = a (t_{1}^{2} + t_{2}^{2}) & \frac{k}{a} = t_{1} + t_{2}$

$∴ 2 h = a [{(t_{1} + t_{2})}^{2} - 2 t_{1} t_{2}]$

$\Rightarrow 2 h = a [{(\frac{k}{a})}^{2} - 2 \cdot (- 4)]$

$\Rightarrow 2 h = \frac{k^{2}}{a} + 8 a$

$\Rightarrow 2 a h = k^{2} + 8 a^{2}$

Replacing $(h, k)$ by $(x, y)$ we get

$y^{2} = 2 a x - 8 a^{2} = 2 a (x - 4 a)$ which is the locus of mid point of $P Q$ .

A variable chord $P Q$ of a parabola $y^{2} = 4 a x$ , subtends a right angle at the vertex. Show that it always passes through a fixed point. Also show that the locus of the point of intersection of the tangents at $P$ and $Q$ is a straight line. Find the locus of the mid point of $P Q$ .

Important Questions on Parabola

Learn and Prepare for any exam you want !

A variable chord PQ of a parabola y2=4ax, subtends a right angle at the vertex. Show that it always passes through a fixed point. Also show that the locus of the point of intersection of the tangents at P and Q is a straight line. Find the locus of the mid point of PQ.

Important Questions on Parabola

Learn and Prepare for any exam you want !

A variable chord $P Q$ of a parabola $y^{2} = 4 a x$ , subtends a right angle at the vertex. Show that it always passes through a fixed point. Also show that the locus of the point of intersection of the tangents at $P$ and $Q$ is a straight line. Find the locus of the mid point of $P Q$ .