Hyperbola

Equation of normal at P(x,y) is Y−y=−‌

dx

dy

(x−x)
Coordinate of G at X axis is (X,0) (let)
⇒X=x+y‌

dy

dx

∴ Co-ordinate of G(x+y‌

dy

dx

,0)
Given distance of G from origin = twice of the abscissa of P.
as distance cannot be −ve, therefore abscissa x should be +ve
On Integrating ⇒‌

y2

2

=‌

x2

2

+c1⇒x2−y2=−2c1
∴ the curve is a hyperbola