Question

The curve for which the length of the normal is equal to the length of the radius vector, are

• A. only circles
• B. only rectangular hyperbolas
• C. either circles or rectangular hyperbolas
• D. None of the above

Answer 1

Answer: (C)

either circles or rectangular hyperbolas

We have length of the normal $=$ radius vector
$\Rightarrow y\sqrt{1+{\left(\frac{dy}{dx}\right)}^2}=\sqrt{x^2+y^2}$
$\Rightarrow y^2\{1+{\left(\frac{dy}{dx}\right)}^2\}=x^2+y^2$
$\Rightarrow y^2{\left(\frac{dy}{dx}\right)}^2=x^2$
$\Rightarrow x=\pm y\frac{dy}{dx}$
$\Rightarrow x=y\frac{dy}{dx}$ or $x=-y\frac{dy}{dx}$
$\Rightarrow xdx-ydy=0$
or $xdx+ydy=0$
$\Rightarrow x^2-y^2=c_1$ or $x^2+y^2=c_2$
Clearly, $x^2-y^2=c_1$ represents a rectangular hyperbola and $x^2+y^2=c_2$ represents circle.