If two tangents to the parabola y^2=8x meet t | vedclass

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

Question

(C) Given parabola is $y^2=8x$. Comparing with $y^2=4ax$,we get $4a=8 \Rightarrow a=2$. Let the parametric coordinates of $P$ and $Q$ be $(at_1^2, 2at

Vedclass · Accepted Answer

$y^2=8(x+2)$

Vedclass · Answer

(C) Given parabola is $y^2=8x$. Comparing with $y^2=4ax$,we get $4a=8 \Rightarrow a=2$. Let the parametric coordinates of $P$ and $Q$ be $(at_1^2, 2at_1)$ and $(at_2^2, 2at_2)$ respectively. So,the equation of the tangent at $P$ is $yt_1=x+at_1^2 \Rightarrow yt_1=x+2t_1^2$ $(i)$. Similarly,the equation of the tangent at $Q$ is $yt_2=x+2t_2^2$ $(ii)$. The tangent at the vertex of the parabola $y^2=8x$ is the $y$-axis,i.e.,$x=0$. To find $M$,we put $x=0$ in equation $(i)$,we get $yt_1=2t_1^2 \Rightarrow y=2t_1$. Thus,$M=(0, 2t_1)$. To find $N$,we put $x=0$ in equation $(ii)$,we get $yt_2=2t_2^2 \Rightarrow y=2t_2$. Thus,$N=(0, 2t_2)$. Given $MN=4$,so $|2t_1-2t_2|=4$ $\Rightarrow |t_1-t_2|=2$ $\Rightarrow (t_1-t_2)^2=4$ $(iii)$. The point of intersection $R(h, k)$ of the tangents at $P$ and $Q$ is given by $(at_1t_2, a(t_1+t_2))$. So,$(h, k) = (2t_1t_2, 2(t_1+t_2))$. This implies $t_1t_2 = h/2$ and $t_1+t_2 = k/2$. We know $(t_1+t_2)^2 = (t_1-t_2)^2 + 4t_1t_2$. Substituting the values,$(k/2)^2 = 4 + 4(h/2)$ $\Rightarrow k^2/4 = 4+2h$ $\Rightarrow k^2 = 16+8h = 8(h+2)$. Replacing $(h, k)$ with $(x, y)$,the locus is $y^2=8(x+2)$.

If two tangents to the parabola $y^2=8x$ meet the tangent at its vertex in $M$ and $N$ such that $MN=4$,then the locus of the point of intersection of those two tangents is

Similar Questions

The condition for which the straight line $y = mx + c$ touches the parabola $y^2 = 4ax$ is

Let $S$ be the set of all $a \in \mathbb{N}$ such that the area of the triangle formed by the tangent at the point $P(b, c)$,where $b, c \in \mathbb{N}$,on the parabola $y^2 = 2ax$ and the lines $x = b$ and $y = 0$ is $16 \text{ unit}^2$. Then $\sum_{a \in S} a$ is equal to $..........$.

The focus of the parabola $y = 2x^{2} + x$ is

$A$ line passes through the point $(-1, 1)$ and makes an angle $\sin^{-1}(\frac{3}{5})$ in the positive direction of the $x$-axis. If this line meets the curve $x^2 = 4y - 9$ at points $A$ and $B$, then the length $|AB|$ is equal to

Two tangents are drawn from a point $(-2, -1)$ to the curve $y^2 = 4x$. If $\alpha$ is the angle between them,then $|\tan \alpha|$ is equal to:

Vedclass Test Series

Exam Paper Generator

Online Exam Module