The normal to a 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 parabola
• B

  a circle
• Correct answer
  C

  a hyperbola
• D

  an ellipse

Solution

equation of normal $Y-y=\left(\frac{\frac{-1}{d y}}{d x}\right) \cdot(X-x) \quad Y=0 \quad \therefore \quad y \frac{d y}{d x}+x=X \quad \therefore \quad a / c \text { to } Q \quad y \cdot \frac{d y}{d x}+x=2 x$ $\mathrm{y} \cdot \frac{\mathrm{dy}}{\mathrm{dx}}=\mathrm{x} \Rightarrow \int \mathrm{y} \mathrm{dy}=\int \mathrm{xdx} \quad \frac{\mathrm{y}^{2}}{2}=\frac{\mathrm{x}^{2}}{2}+\mathrm{c} \quad \frac{\mathrm{y}^{2}-\mathrm{x}^{2}}{2}=\mathrm{c}$ a hyperbola

Hence, the correct answer is option: 'C'.