Question

If the angle between the curves $y^2=16x$ and $9x^2+b{y}^2=16$ is $\frac{\pi }{2}$, then the value of $b$ is

• A. $2$
• B. $4$
• C. $\frac{9}{2}$
• D. $\frac{13}{4}$

Answer 1

Answer: (C)

$\frac{9}{2}$

Given: Equation of the curves $y^2=16x$ ...(i)

and $9x^2+{by}^2=16$ ...(ii)

Angle between the curves (i) and (ii) $=\frac{\pi }{2}$.

Differentiating Eq. (i), we get

$2y\frac{dy}{dx}=16$ or $\frac{dy}{dx}=\frac{8}{y}=m_1$

and differentiating Eq. (ii), we get

$18x+2by\frac{dy}{dx}=0$ or $by\frac{dy}{dx}=-9x$

or $\frac{dy}{dx}=\frac{-9x}{by}=m_2$

We know that $m_1\times m_2=-1$

or $\left(\frac{8}{y}\right)\times \left(\frac{-9x}{by}\right)=-1$ or ${by}^2=72x$

or $16bx=72x$

or $b=\frac{9}{2}$.