Question

If the curves $y^2=4ax$ and $xy=c^2$ cut orthogonally then $\frac{c^4}{a^4}=$

• A. $4$
• B. $8$
• C. $16$
• D. $32$

Answer 1

The correct option is D $32$

$y^2=4ax,xy=c^2$ cut orthogonally

Let they intersect at $(x_1,y_1)$

$y^2=4ax$

$2y\frac{dy}{dx}=4a$

$\frac{dy}{dx}=\frac{2a}{y}$

$\frac{dy}{dx}{|}_{(x_1,y_1)}=\frac{2a}{y_1}$ …..$\left(1\right)$

$xy=c^2$

$y+x\frac{dy}{dx}=0$

$\frac{dy}{dx}{|}_{(x_1,y_1)}=\frac{-y_1}{x_1}$ ……$\left(2\right)$

From $\left(1\right),\left(2\right)$ $(\because m_1{m}_2=-1)$

$\frac{2a}{\text{/}{y}_1}\times \frac{\text{/}-\text{/}{y}_1}{x_1}=\text{/}-1$

$x_1=2a$

From $y^2=4ax$

$y_1=\sqrt{8a^2}=2\sqrt{2}a$

$xy=c^2$

$x_1{y}_1=c^2$

$2a\left(2\sqrt{2}a\right)=c^2$

$\frac{c^2}{a^2}=4\sqrt{2}$

$\frac{c^4}{a^4}=32$.