Question

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

Answer 1

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