Question

Tangents are drawn from different points on the line $x-y+10=0$ to the parabola $y^2=4x$. If the chords of contact pass through a fixed point, then the coordinates of the fixed point is

• A. $(2,5)$
• B. $(5,2)$
• C. $(10,2)$
• D. $(2,10)$

Answer 1

The correct option is C $(10,2)$

Let any point on the line
$x-y+10=0$ be $(t,t+10)$
From this point, the equation of chord of contact to the parabola $y^2=4x$ is,
$T=0$
$\Rightarrow y(t+10)=2(x+t)$
$\Rightarrow 2x-10y+t(2-y)=0$
So, the fixed point is the intersection of line,
$\Rightarrow 2x-10y=0\&2-y=0$
$\Rightarrow y=2\Rightarrow x=10$

Hence, the fixed point is $(10,2)$