Question

A normal is drawn at a point $S(x,y)$ of curve, meet the x-axis at a point $R$. If $SR$ is a constant length $\lambda$. If the curve passing through $(0,\lambda )$ then its equation is

• A. A circle with radius $\lambda$
• B. An ellipse
• C. A hyperbola
• D. A parabola

Answer 1

Answer: (A)

A circle with radius $\lambda$

Equation of normal at $S(x,y)$
$Y-y=-\frac{dx}{dy}(X-x)$
so co-ordinates of $R(x+y\frac{dy}{dx},0)$
$\therefore {\left(SR\right)}^2={(X-x)}^2+{(0-y)}^2={\left(y\frac{dy}{dx}\right)}^2+y^2$
$\Rightarrow {\lambda }^2={\left(y\frac{dy}{dx}\right)}^2+y^2\Rightarrow y\frac{dy}{dx}=\pm \sqrt{{\lambda }^2-y^2}$
$\Rightarrow \frac{ydy}{\sqrt{{\lambda }^2-y^2}}=\pm dx$
$\Rightarrow -\sqrt{{\lambda }^2-y^2}=\pm x+\alpha ,(\alpha \in R)...(\ast )$
using $(0,\lambda )$ to determine $\alpha$. $\therefore$ $\alpha =0$
so equation $(\ast )$ gives
$x^2+y^2={\lambda }^2$ which represents a circle whose radius is $\lambda$