First slide

Definition of a circle

Question

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

Moderate

A

$(25, - 50)$
B

$(- 25, 50)$
C

$(- 25, - 50)$
D

$(25, 50)$

Solution

Let the required point be $R (x_{1}, y_{1})$ Then, $P Q$
is the chord of contact of tangents drawn from $R (x_{1}, y_{1})$ to
$x^{2} + y^{2} = 25$ So, the equation of $P Q$ is $x x_{1} + y y_{1} = 25$
This equation and $x - 2 y + 1 = 0$ represent the same line.
$∴ \frac{x_{1}}{1} = \frac{y_{1}}{- 2} = \frac{- 25}{1} \Rightarrow x_{1} = - 25$ and $y = 50$
Hence, the required point is $(- 25, 50)$

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