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

  • Option 1)

    circle

  • Option 2)

    hyperbola

  • Option 3)

    ellipse

  • Option 4)

    parabola

 

Answers (1)

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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