The normal to a curve at `P(x , y)` meet the x-axis at `Gdot` If the distance of `G` from the origin is twice the abscissa of `P` , then the curve is a (a) parabola (b) circle (c) hyperbola (d) ellipse

Options

A. parabola
B. circle
C. hyperbola
D. ellipse

Answer

C

Explanation

Slope of tangent =`(dy)/(dx)`
`therefore` Slope of normal `=-(dx)/(dy)`
Thus, the equation of normal is
`Y-y=-(dx)/(dy)(X-x)`
This meets x-axis, where
`-y=-(dx)/(dy)(X-x)` or `X=x+y(dy)/(dx)`
`therefore` G is `(x+y(dy)/(dx),0)`
`therefore` OG =2x
`therefore x+y(dy)/(dx) =2x`
or `y(dy)/(dx)=x` or `ydy=xdx`
Integrating, we get `y^(2)/2=x^(2)/2+C/2`
or `y^(2)-x^(2)=C`, which is a hyperbola.