Question

The equation of the family of curves which inetersect the hyperbola $xy=2$ orthogonally is:

• A. $y=\frac{x^3}{6}+c$
• B. $y=\frac{x^2}{4}+c$
• C. $y=-\frac{x^3}{6}+c$
• D. $y=-\frac{x^2}{4}+c$

Answer 1

The correct option is A $y=\frac{x^3}{6}+c$

Let $m_1=\frac{dy}{dx}$ for required family of curves at $(x,y)$.

Let $m_2=\frac{dy}{dx}$ for the hyperbola $xy=2$.

Then $m_2=\frac{dy}{dx}=-\frac{2}{x^2}$.

Since the required family of curves is orthogonal to the hyperbola,

$\therefore m_1\times m_2=-1$

$\Rightarrow \frac{dy}{dx}\times (-\frac{2}{x^2})=-1\Rightarrow \frac{dy}{dx}=\frac{x^2}{2}\Rightarrow dy=\frac{x^2}{2}dx$

Integrating, we get $y=\frac{x^3}{6}+c$, which is the required family.