If a normal chord of the parabola y^2 = 4ax s | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(B) Let the coordinates of the points $P$ and $Q$ on the parabola $y^2 = 4ax$ be $P(at_1^2, 2at_1)$ and $Q(at_2^2, 2at_2)$.Since $PQ$ is a normal chor

Vedclass · Accepted Answer

$\pm \sqrt{2}$

Vedclass · Answer

(B) Let the coordinates of the points $P$ and $Q$ on the parabola $y^2 = 4ax$ be $P(at_1^2, 2at_1)$ and $Q(at_2^2, 2at_2)$.Since $PQ$ is a normal chord,we have the relation $t_2 = -t_1 - \frac{2}{t_1} \quad \dots(1)$.The slopes of $OP$ and $OQ$ are $m_1 = \frac{2at_1}{at_1^2} = \frac{2}{t_1}$ and $m_2 = \frac{2at_2}{at_2^2} = \frac{2}{t_2}$ respectively.Since $PQ$ subtends a right angle at the vertex $O(0,0)$,the product of the slopes of $OP$ and $OQ$ is $-1$.$\frac{2}{t_1} 	imes \frac{2}{t_2} = -1 \Rightarrow t_1t_2 = -4 \quad \dots(2)$.Substituting $t_2$ from $(1)$ into $(2)$,we get $t_1(-t_1 - \frac{2}{t_1}) = -4$.$-t_1^2 - 2 = -4$ $\Rightarrow t_1^2 = 2$ $\Rightarrow t_1 = \pm \sqrt{2}$.The slope of the normal at $P$ is $m = -t_1$.Therefore,the slope of the normal chord is $m = \mp \sqrt{2}$.Since the options represent the set of possible slopes,the correct choice is $\pm \sqrt{2}$.

If a normal chord of the parabola $y^2 = 4ax$ subtends a right angle at the vertex,its slope is-

Similar Questions

If the focus divides a focal chord of the parabola $y^2 = 16x$ into $2$ parts having lengths $a$ and $c$,such that $a, b, c$ are in $H.P.$,then the value of $b$ is equal to:

Find the length of the chord intercepted by the line $4y = 3x - 48$ on the parabola $y^{2} = 64x$.

$A$ normal chord of the parabola $y^2 = 4x$ subtending a right angle at the vertex makes an acute angle $\theta$ with the $x$-axis. Then $\theta$ is equal to:

The points of intersection of the parabolas $y^2=5x$ and $x^2=5y$ lie on the line

What is the common tangent to the parabola $y^{2} = 8ax$ and the circle $x^{2} + y^{2} = 2a^{2}$?

Vedclass Test Series

Exam Paper Generator

Online Exam Module