Question

A normal is drawn at a point $P(x,y)$ of a curve, It meets the $x$-axis at $Q$. If $PQ$ is of constant length $k$. Such a curve passing through $(0,k)$ is :

• A. a circle with centre $(0,0)$
• B. a hyperbola with eccentricity $\sqrt{2}$
• C. $x^2+y^2=k^2$
• D. $x^2-y^2=k^2$

Answer 1

Answer: (A), (C)

Equation of normal at $P(x,y)$ is
$Y-y=-\frac{dx}{dy}(X-x)$
so coordinate of $Q$ is $\left(x+y\frac{dy}{dx},0\right)$
Thus $(PQ{)}^2=(X-x{)}^2+(0-y{)}^2$
$\Rightarrow k^2={\left(y\frac{dy}{dx}\right)}^2+y^2$
$\Rightarrow y\frac{dy}{dx}=\pm \sqrt{k^2-y^2}$
$\Rightarrow \frac{ydy}{\sqrt{k^2-y^2}}=\pm \int dx$
$\Rightarrow =\pm x+c$
$\Rightarrow$ it passes through $(0,k)$
$\Rightarrow c=0\Rightarrow -\sqrt{k^2-y^2}=\pm x$
$k^2-y^2=x^2$
$x^2+y^2=k^2$