First slide

Definition of a circle

Question

Let the tangents drawn from the origin to the circle, $x^{2} + y^{2} - 8 x - 4 y + 16 = 0$ touch it at the points A and B. Then ${(A B)}^{2}$ is equal to:

Moderate

A

$\frac{64}{5}$
B

$\frac{56}{5}$
C

$\frac{52}{5}$
D

$\frac{32}{5}$

Solution

$L e n g t h o f \tan g e n t = L = \sqrt{S_{1}} = \sqrt{16} = 4$
$R a d i u s o f t h e c i r c l e = R = \sqrt{16 + 4 - 16} = 2$
$∴$ Length of Chord of contact $= \frac{2 L R}{\sqrt{L^{2} + R^{2}}} = \frac{2 \times 4 \times 2}{\sqrt{16 + 4}} = \frac{16}{\sqrt{20}}$
$\Rightarrow$ Square of length of chord of contact $= \frac{64}{5}$

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