Question

The normal to the curve at $P(x,y)$ meets the $x$-axis at $G$. If the distance of $G$ from the origin is twice the abscissa of $P$, then the curve is a

• A. Ellipse
• B. Parabola
• C. Circle
• D. Hyperbola

Answer 1

The correct option is D

Hyperbola

Explanation for the correct option:

Find the shape of curve based on given information

Equation for the normal is given as

$Y-y=-\frac{dx}{dy}(X-x)$

The intercept of the normal on the $x$-axis is given by

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

Hence the intercept is given as $\left(x+y\frac{dy}{dx},0\right)$

According to the given condition

$x+y\frac{dy}{dx}=2x$

$\Rightarrow$ $ydy=xdx$

Integrating both sides

$\int ydy=\int xdx$

We know that,

$\mathbf{\int }{\mathbf{x}}^{\mathbf{n}}\mathbf{dx}\mathbf{}\mathbf{}\mathbf{=}\mathbf{}\frac{{\mathbf{x}}^{\mathbf{n}\mathbf{+}\mathbf{1}}}{\mathbf{n}\mathbf{+}\mathbf{1}}\mathbf{+}\mathbf{k}$ where $k$ is constant of integration

Substitute $n=1$

$\Rightarrow$ $\int xdx=\frac{x^2}{2}+c$

Similarly,

$\Rightarrow$ $\int ydy=\frac{y^2}{2}+k_1$

$\Rightarrow$ $\frac{y^2}{2}=\frac{x^2}{2}+c_1$

$\Rightarrow$ $x^2-y^2=c$

Therefore, the curve is a rectangular hyperbola.

Hence, option$\left(D\right)$ is the correct answer.