First slide

Definition of a circle

Question

If the straight line $x - 2 y + 1 = 0$ intersects the circle $x^{2} + y^{2} = 25$ in points $P$ and $Q$ , then the coordinates of the point of intersection of tangents drawn at $P$ and $Q$ to the circle $x^{2} + y^{2} = 25$ are

Moderate

A

$(25, 50)$
B

$(- 25, - 50)$
C

$(- 25, 50)$
D

$(25, - 50)$

Solution

Let $R (h, k)$ be the point of intersection of tangents drawn at $P$ and $Q$ to the given circle. Then, $P Q$ is the chord of contact of tangents drawn from $R$ to $x^{2} + y^{2} = 25$ So its equation is
$h x + k y - 25 = 0 .$ …(i)
It is given that the equation of $P Q$ is
$x - 2 y + 1 = 0$ …(ii)
Since (i) and (ii) represent the same line.
$∴ \frac{h}{1} = \frac{k}{- 2} = \frac{- 25}{1} \Rightarrow h = - 25, k = 50$
Hence, the required point is $(- 25, 50) .$

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