First slide

Definition of a circle

Question

$A B is a chord of the circle x^{2} + y^{2} = 25 . The tangents to the circle at A and B intersect$ $at C . If (2, 3) is the midpoint of A B, then the area of quadrilateral O A C B (Where O is$ $origin) is$

Moderate

A

$\frac{50}{\sqrt{3}}$
B

$50 \sqrt{\frac{3}{13}}$
C

$50 \sqrt{3}$
D

$\frac{50}{\sqrt{13}}$

Solution

$\begin{array}{l} Here O B = O A = 5 ( radius of circle) \\ O P = \sqrt{(2 - 0)^{2} + (3 - 0)^{2}} = \sqrt{13} \\ Let ∠ O B P = θ, then ∠ P C B = θ \\ From Δ O P B, \cos (90 - θ) = \frac{O P}{O B} \\ = \frac{\sqrt{13}}{5} \\ \Rightarrow \sin θ = \frac{\sqrt{13}}{5} \\ \Rightarrow \cot θ = \frac{2 \sqrt{3}}{\sqrt{13}} \end{array}$
$Area of quadrilateral O A C B = 2 \times \frac{1}{2} \times O B \times B C$
$\begin{array}{l} From Δ O B C, \cot θ = \frac{B C}{O B} = \frac{B C}{5} \\ ∴ Area of quadrilateral O A C B \end{array}$
$\begin{array}{l} = 5 \times 5 \cot θ \\ = 25 \times \frac{2 \sqrt{3}}{\sqrt{13}} \\ = 50 \sqrt{\frac{3}{13}} sq.units \end{array}$

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