Question

The lengths of tangents from two given points to a given circle are $t_1\&t_2$. If the two points are conjugate to each other w.r.t. the given circle, then the distance between the points will be

• A. $\sqrt{\frac{t_1^2+t_2^2}{2}}$
• B. $\sqrt{t_1^2+t_2^2}$
• C. $\sqrt{t_1{t}_2}$
• D. $\frac{2t_1{t}_2}{t_1+t_2}$

Answer 1

Answer: (B)

$\sqrt{t_1^2+t_2^2}$

Let P and Q be the conjugate points and O be the center.

Since P, Q are conjugate

$⟹P{Q}^2=O{P}^2+O{Q}^2–2r^2$

$⟹P{Q}^2=(t_1^2+r^2)+(t_2^2+r^2)–2r^2$

$⟹P{Q}^2=t_1^2+t_2^2$

$⟹PQ=\sqrt{t_1^2+t_2^2}$