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Question

The normal to the 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


A

Ellipse

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B

Parabola

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C

Circle

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D

Hyperbola

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Solution

The correct option is D

Hyperbola


Explanation for the correct option:

Find the shape of curve based on given information

Equation for the normal is given as

Y-y=-dxdy(X-x)

The intercept of the normal on the x-axis is given by

ydydx+x=X

Hence the intercept is given as x+ydydx,0

According to the given condition

x+ydydx=2x

ydy=xdx

Integrating both sides

ydy=xdx

We know that,

xndx=xn+1n+1+k where k is constant of integration

Substitute n=1

xdx=x22+c

Similarly,

ydy=y22+k1

y22=x22+c1

x2-y2=c

Therefore, the curve is a rectangular hyperbola.

Hence, optionD is the correct answer.


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