Question

Tangents are drawn to $x^2+y^2=1$ from any arbitrary point $P$ on the line $2x+y-4=0$. The corresponding chord of contact passes through a fixed point whose coordinates are

• A. $(\frac{1}{4},\frac{1}{2})$
• B. $(\frac{1}{2},1)$
• C. $(\frac{1}{2},\frac{1}{4})$
• D. $(1,\frac{1}{2})$

Answer 1

The correct option is C $(\frac{1}{2},\frac{1}{4})$

Let $P\equiv (a,4-2a)$ be any point on $2x+y=4$
Equation of chord of contact drawn from $P$ is
$T=0\Rightarrow x\cdot a+y.(4-2a)=1$
$\Rightarrow (4y-1)+a(x-2y)=0$
the above equation represents family of lines which passes through intersection points of $y=1/4$ and $x=2y$

So, the required point is $(\frac{1}{2},\frac{1}{4})$