Let P( a _ 1 , b _ 1 ) and Q( a _ 2 , b _ 2 ) be two distinct...

Given,

$P\left({a}_{1}, {b}_{1}\right)$ and $Q\left({a}_{2}, {b}_{2}\right)$ be two distinct points on a circle with center $C ( \sqrt{2}, \sqrt{3} )$,

And $O$ be the origin and $OC$ be perpendicular to both $CP$ and $CQ$,

So, $PCQ$ will be a straight line

Now on plotting the diagram we get,

So, $OC = \sqrt{{\left(\sqrt{2}\right)}^{2} + {\left(\sqrt{3}\right)}^{2}} = \sqrt{5}$

Now let $CP = CQ = I$
Now by using area of triangle $OCP = \dfrac{1}{2} \times OC \times I$ we get,

$\dfrac{\sqrt{35}}{2} = \dfrac{1}{2} \times \sqrt{5} \times I ⇒ I = \sqrt{7}$

And $OP = OQ = \sqrt{O{C}^{2} + {I}^{2}} = \sqrt{12}$

So, ${a}_{1}^{2} + {b}_{1}^{2} + {a}_{2}^{2} + {b}_{2}^{2} = O{P}^{2} + O{Q}^{2} = 12 + 12 = 24$