QJEEM23Ma0293: JEE Main 2023 Mathematics - Class 11 JEEM23Ma0293 Solution: Let and be two distinct points on a circle with center . Let be …

Question

Verified Solution • 2025–26 Session moderate

Let

P\left(a_1, b_1\right)

and

Q\left(a_2, b_2\right)

be two distinct points on a circle with center

C(\sqrt{2}, \sqrt{3})

. Let

\mathrm{O}

be the origin and

\mathrm{OC}

be perpendicular to both

\mathrm{CP}

and

\mathrm{CQ}

. If the area of the triangle

\mathrm{OCP}

is

\frac{\sqrt{35}}{2}

, then

a_1^2+a_2^2+b_1^2+b_2^2

is equal to :

Accepted Answer

$OC \,\bot \,CP$ and $OC \,\bot \, CQ$

$\Rightarrow PCQ$ is a straight line

JEE Main 2023 (Online) 30th January Evening Shift Mathematics - Circle Question 43 English Explanation

$OC = \sqrt {{{(\sqrt 2 )}^2} + {{(\sqrt 3 )}^2}} = \sqrt 5$

Let $CP = CQ = I$

$[OCP] = {1 \over 2} \times OC \times I = {{\sqrt {35} } \over 2}$

$I = \sqrt 7$

$OP = OQ = \sqrt {{{(OC)}^2} + {I^2}} = \sqrt {5 + 7} = \sqrt {12}$

$a_1^2 + a_2^2 + b_1^2 + b_2^2 = \left( {a_1^2 + b_2^2} \right) + \left( {a_2^2 + b_2^2} \right)$

$O{P^2} + O{Q^2} = 12 + 12 = 24$

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