Let P ( a 1 , b 1 ) and Q ( a 2 , b 2 ) be two distinct points on a circle with center C ( 2 , 3 ) . Let O be the origin and OC be perpendicular to both CP and CQ. If the area of the triangle OCP is 35 2 , then a 1 2 + a 2 2 + b 1 2 + b 2 2 is equ

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Solution

Q.
Let $P (a_{1}, b_{1})$ and $Q (a_{2}, b_{2})$ be two distinct points on a circle with center $C (\sqrt{2}, \sqrt{3})$ . Let O be the origin and OC be perpendicular to both CP and CQ. If the area of the triangle OCP is $\frac{\sqrt{35}}{2}$ , then $a_{1}^{2} + a_{2}^{2} + b_{1}^{2} + b_{2}^{2}$ is equal to ________ . [2023]

Ans.
(24)

$Area of ∆ O C P = \frac{\sqrt{35}}{2}$

$\frac{1}{2} \times P C \times O C = \frac{\sqrt{35}}{2}$

$\frac{1}{2} \times P C \times \sqrt{5} = \frac{\sqrt{35}}{2}$

$P C = \sqrt{7}$

$Now, O Q^{2} = 5 + 7 = 12$

$O P^{2} = 5 + 7 = 12$

$a_{1}^{2} + b_{1}^{2} = O P^{2} and a_{2}^{2} + b_{2}^{2} = O Q^{2}$

$∴$ $a_{1}^{2} + a_{2}^{2} + b_{1}^{2} + b_{2}^{2} = O P^{2} + O Q^{2} = 12 + 12 = 24$

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