Question

If the curves $2x = y^2$ and $2xy = K$ intersect perpendicularly, then the value of $K^2$ is

• A. $4$
• B. $2\sqrt{2}$
• C. $2$
• D. $8$

Answer 1

Answer: (D)

$8$

Answer 2

$2 x=y^{2} \ldots(1)$
$2 x y=K \ldots(2)$
Solving $(1)$ and $(2)$, we get
$( x , y )=\left( K ^{(2 / 3)} / 2, K ^{(1 / 3)}\right)$
Differentiating $(1)$ and $(2)$ w.r.t. $x$
$m _{1}= dy / dx =1 / y \ldots(3)$
$m_{2}=d y / d x=-y / x \ldots(4)$
Both curves intersect each other perpendicularly
$\therefore m _{1} m _{2}=-1$
$\Rightarrow-1 / x=-1$
$\Rightarrow x =1$
$\Rightarrow K ^{(2 / 3)}=2$
$\Rightarrow K ^{2}=8$