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Q.
Tangents are drawn from the point $(-1,2)$ on the parabola $y ^2=4 x$. The length, these tangents will intercept on the line $x =2$ :
Conic Sections
Solution:
Let slope of tangent be $m$ So equation of tangent is
$y=m x+\frac{1}{m}$
Now tangent passes through $(-1,2)$ so
$\Rightarrow m^2+2 m-1=0 $
$\Rightarrow m=-1 \pm \sqrt{2}$
equation of tangents are
$y=(-1+\sqrt{2}) x+\frac{1}{-1+\sqrt{2}}$....(i)
$y=(-1-\sqrt{2}) x-\frac{1}{1+\sqrt{2}}$ ...(ii)
intercept of tangent (i) & (ii) on line $x =2$ is
$y_1=3 \sqrt{2}-1$ & $y_2=-3 \sqrt{2}-1$ respectively.
Now $y_1-y_2$ is $6 \sqrt{2}$