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Question

Two parabola y2=4a(xλ1), and x2=4a(yλ2) always touch each other, where λ1 and λ2 being variable parameters. Then their points of contact lie on a


A

Circle

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B

Hyperbola

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C

Ellipse

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D

Parabola

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Solution

The correct option is B

Hyperbola


Since the parabolas touch each other, they will have the common tangent.

Finding dydx:

y2=4a(xλ1)

Differentiating with respect to x, we get

2ydydx=4a ... (1)

dydx=4a2y=2ay

Differentiating x2=4a(yλ2) with respect to y, we get

2xdxdy=4a

dydx=2x4a=x2a ... (2)

Equating equations (1) and (2), we get

2ay=x2a

xy=4a2 which is a hyperbola.

Hence, their points of contact lie on a hyperbola.


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