JEE 2023 — Mathematics PYQ

Equation of tangent $y-f(b)=f^{\prime }(b)(x-b)$
Which passes through $(0,c)$
$\Rightarrow c-f(b)=f^{\prime }(b)(-b)$
$\Rightarrow \frac{3}{b}-f(b)=f^{\prime }(b)(-b)$
$\Rightarrow \frac{b{f}^{\prime }(b)-f(b)}{b^2}=\frac{-3}{b^3}$
$\Rightarrow d\left(\frac{f(b)}{b}\right)=\frac{-3}{b^3}\Rightarrow \frac{f(b)}{b}=\frac{3}{2b^2}+\lambda$
passes through $\left(1,\frac{3}{2}\right)$
$\Rightarrow \frac{3}{2}=\frac{3}{2}+\lambda \Rightarrow \lambda =0$
$f(b)=\frac{3}{2b}$

$f(a)=\frac{1}{2}\Rightarrow a=3$
$\Rightarrow Q\left(3,\frac{1}{2}\right)$
$P{Q}^2=(3-1{)}^2+{\left(\frac{1}{2}-\frac{3}{2}\right)}^2$
$P{Q}^2=2^2+(-1{)}^2=4+1=5$