# The normal to a curve at $\dpi{100} P(x,y)$ meets the $\dpi{100} x-$axis at $\dpi{100} G$ . If the distance of  $\dpi{100} G$ from the origin is twice the abscissa of $\dpi{100} P$ , then the curve is a Option 1) circle Option 2) hyperbola Option 3) ellipse Option 4) parabola

D Divya Saini

As we learnt in

Slope of a line -

$m= \frac{y_{2}-y_{1}}{x_{2}-x_{1}}$

- wherein

Slope of line joining A(x1,y1) and  B(x2,y2) .

If slope of tangent = $m$

We get  $\frac{y-y_{1}}{x-x_{1}}=\frac{-1}{m}$    as equation of normal.

If normal meets x-axis again, we get:

$x=my_{1}+x_{1}$

Distance of QG = 2 (abscissa of P)

i.e.,    $my_{1}+x_{1}=2x_{1}$

$\Rightarrow m=\frac{x_{1}}{y_{1}}\:\:\:\:at\:\:\:\:(x_{1}y_{1})$

Generalising  $\frac{dy}{dx}=\frac{x}{y}$

$\frac{y^{2}}{2}=\frac{x^{2}}{2}+c$

$\frac{x^{2}}{2}-\frac{y^{2}}{2}=c$    which is a Hyperbola.

Option 1)

circle

This option is incorrect.

Option 2)

hyperbola

This option is correct.

Option 3)

ellipse

This option is incorrect.

Option 4)

parabola

This option is incorrect.

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