Question

Pair of tangents are drawn from every point on the line $3x+4y=12$ on the circle $x^2+y^2=4$. Their variable chord of contact always passes through a fixed point whose co-ordinates are-

• A. $(\frac{4}{3},\frac{3}{4})$
• B. $(\frac{3}{4},\frac{3}{4})$
• C. $(1,1)$
• D. $(1,\frac{4}{3})$

Answer 1

The correct option is D $(1,\frac{4}{3})$

Let $P(x_1,y_1)$ be a point on the line $3x+4y=12$
Equation of variable chord of contact of $P(x_1,y_1)$ w.r.t circle $x^2+y^2=4$ (figure)
$x{x}_1+y{y}_1-4=0$ ...(1)

Also $3x_1+4y_1-12=0$$\Rightarrow x_1+\frac{4}{3}{y}_1-4=0$ ...(2)
Comparing $\left(1\right)$ and $\left(2\right)$, we get
$x=1;y=\frac{4}{3}$
$\therefore$ Variable chord of contact always passes through $(1,\frac{4}{3})$