Question

Let $OAB$ be a triangle where $A\equiv (\sqrt{2},p+1),B\equiv \left(\sqrt{2},p^2+3p+4\right)$ and ' $O$ ' is the origin. If orthocentre of the $△OAB$ is $H$ and least value of area of $\Delta$ formed by joining the points, orthocentres of $△OAH$ and $△OBH$ and the origin ' $O$ ' is $S$ then find the value of $S^2$.

Answer 1

Answer: 2

$A\equiv (\sqrt{2},p+1),B=\left(\sqrt{2},p^2+3p+4\right)$ and orthocenter of $△OAH\equiv B$
orthocenter of $△OBH\equiv A$
So, $S=$ Area of $OAB$
$S=\frac{1}{2}\times \left(p^2+3p+4-p-1\right)\times \sqrt{2}$
$S=\frac{p^2+2p+3}{\sqrt{2}}\Rightarrow S_{\min }=\frac{(p+1{)}^2+2}{\sqrt{2}}$
$S_{\min }=\sqrt{2}\Rightarrow S^2=2$