Question

Tangents are drawn from the point $(-1,2)$ to the parabola $y^2=4x$. The length of the intercept made by the line $x=2$ on these tangents is

• A. 6
• B.
• C.
• D. none

Answer 1

The correct option is B

If the line $x=2$ intersects these tangents at $(x_1,y_1)$ and $(x_2,y_2)$ then the length of the intercept is given by $|y_1-y_2|$
$S{S}_1=T^2$ is the equation of pair of tangents
$\Rightarrow (y^2-4x)\left(8\right)=4(y-x+1{)}^2$
$\Rightarrow y^2-2y(1-x)-(x^2+6x+1)=0$
Put $x=2$
$\Rightarrow y^2+2y-17=0$

Clearly, $y_1,y_2$ are the roots of the equation $y^2+2y-17=0$

$\Rightarrow y_1+y_2=-2$

$y_1\cdot y_2=-17$

Now, $|y_1-y_2|=\sqrt{(y_1+y_2{)}^2-4y_1\cdot y_2}$

$=\sqrt{2^2-4(-17)}$

$=\sqrt{72}$

$=6\sqrt{2}$
$\therefore |y_1-y_2|=6\sqrt{2}$