JEE Main · 2007 · 29 Apr 2007 · Offline · Differential Equations

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

Answer: (A), (D)

Solution

The normal at

P (x, y)

meets the

x

-axis at

G = (x + y \frac{d y}{d x}, 0)

, and the distance of

G

from the origin being twice the abscissa means

x + y \frac{d y}{d x} = 2∣ x ∣

. One sign gives

y \frac{d y}{d x} = x

, which integrates to

x^{2} - y^{2} = constant

, a rectangular hyperbola. The other sign gives

y \frac{d y}{d x} = - 3 x

, which integrates to

3 x^{2} + y^{2} = constant

, an ellipse. So the curve may be either an ellipse or a hyperbola.

JEE Main · 2007 · 29 Apr 2007 · Offline · One or more correct options

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