Question

Tangents are drawn from each point on the line $2x+y=4$ to the circle $x^2+y^2=1$.The chords of contact pass through the point

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

Answer 1

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

Given: $2x+y=4$
Let $x=\alpha \Rightarrow y=4-2\alpha$
The chord of contact of the tangents from the point $(\alpha ,4-2\alpha )$ is $S_1=0$
$\alpha x+(4-2\alpha )y-1=0,(x-2y)+4y-1=0$
The lines are concurrent at the point of intersection of $x-2y=0$ and $4y-1=0$
On solving we get $4y=1$ or $y=\frac{1}{4}$
and $x-2y=0$ or $x=2y=2\times \frac{1}{4}=\frac{1}{2}$
Thus, the point of intersection is at $(\frac{1}{2},\frac{1}{4})$