Multiple choice

The Chord of contact of tangents from a point $P$ to a circle passes through $Q$. If $l_1$ and $l_2$ are the lengths of the tangents from $P$ and $Q$ to the circle, then $PQ$ is equal to

  1. $\displaystyle \frac{l_1+l_2} {2}$
  2. $\displaystyle \frac{l_1-l_2} {2}$
  3. $\displaystyle \sqrt{l^2_1+l^2_2}$
  4. $\displaystyle {2}\sqrt{l^2_1+l^2_2}$
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C Correct answer
Explanation

For a point P outside a circle, the length of the tangent is sqrt(S1). The chord of contact from P is the line polar of P. Using properties of poles and polars, the relationship between the lengths of tangents and the distance between points is derived as PQ = sqrt(l1^2 + l2^2).

AI explanation

Let the centre of the circle be the origin O. The polar of P is the chord of contact, and since it passes through Q, Q lies on the polar of P, implying P lies on the polar of Q by the pole and polar relationship. This means OP and OQ are perpendicular, making triangle POQ a right triangle. Using the Pythagorean theorem, the square of PQ equals the square of the tangent length from P plus the square of the tangent length from Q, so PQ equals the square root of l1 squared plus l2 squared.

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