Q53 / 120 · JEE Main 2007 · Mathematics

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

Correct answer
A
ellipse
B
parabola
C
circle
D
pair of straight lines

Solution

Equation of normal is $Y-y=-\frac{d x}{d y}(X-x)$ $\begin{aligned} & \Rightarrow G \equiv\left(x+y \frac{d y}{d x}, 0\right) \\ & \left|x+y \frac{d y}{d x}\right|=|2 x| \\ & \Rightarrow y \frac{d y}{d x}=x \text { or } y \frac{d y}{d x}=-3 x \\ & y d y=x d x \text { or } y d y=-3 x d x \\ & \frac{y^2}{2}=\frac{x^2}{2}+c \text { or } \frac{y^2}{2}=-\frac{3 x^2}{2}+c \\ & x^2-y^2=-2 c \text { or } 3 x^2+y^2=2 c. \end{aligned}$

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

Relevant Questions

The shortest distance between the curves and is :03 Apr, 2025 (E)Tangent Normal
Let the quadratic curve passing through the point - 1 , 0 and touching the line y = x at 1 , 1 be y = f x . Then the x -intercept of the nor10 Apr, 2023 (E)Tangent Normal
Let the tangent to the curve x 2 + 2 x - 4 y + 9 = 0 at the point P 1 , 3 on it meet the y - axis at A . Let the line passing through P and 06 Apr, 2023 (M)Tangent Normal
Let a curve y = f x , x 0 , pass through the points P 1 , 3 2 and Q a , 1 2 . If the tangent at any point R ( b , f ( b ) ) to the given cur06 Apr, 2023 (E)Tangent Normal
The number of points on the curve y = 54 x 5 - 135 x 4 - 70 x 3 + 180 x 2 + 210 x at which the normal lines are parallel to x + 90 y + 2 = 030 Jan, 2023 (M)Tangent Normal
If the equation of the normal to the curve y = x - a x + b x - 2 at the point 1 , - 3 is x - 4 y = 13 then the value of a + b is equal to __29 Jan, 2023 (E)Tangent Normal