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 (1) ellipse (2) parabola (3) circle (4) hyperbola

Options

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

Answer

B, C

Explanation

Normal at `P(x_(1)y_(1))`
`y-y_(1)=(-1)/m(x-k_(1))`
Where `y=0, x=x_(1)+my_(1)`
We have `y=0, x=x_(1)+my_(1)`
We hve `|x_(1)+my_(1)|=2|x_(1)|`
`implies x_(1)+my_(1)=+-2x_(1)`
Case I: `x_(1)+my_(1)=2x_(1)impliesm=(x_(1))/(y_(1))`
`implies(dy)/(dt)=x/y( :' m= ((dy)/(dx))_("at"(x_(1)y_(1))))`
Case -II: `x_(1)+my_(1)=-2x_(1),m=(-3x_(1))/(y_(1))`
`implies x dx - h dxy =0, (x^(2))/2-(y^(2))/2=C-a` hyperbola
Case II: `x_(1)my_(1)=-2x_(1),m=(-3x_(1))/(y_(1))`
`y (dy)/(dx)+3x=0`
`implies(y^(2))/2+(3x^(2))/2=C` (Ellipse)