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Question

The locus of a point P(α,β) moving under the condition that the line y = αx + β is a tangent to the hyperbola x2a2y2b2=1.


A

a hyperbola

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B

a parabola

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C

a circle

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D

an ellipse

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Solution

The correct option is A

a hyperbola


Here we are dealing with a locus of moving point α,β.

Given that α,β satisfies the condition that y=αx+β is a tangent to hyperbola x2a2y2b2=1

We known the general form of tangent of slope 'm' which is y=±a2m2b2

Since y=αx+β ia a tangent, we can compare the coefficients of the the 2 equations and equate them if

coefficient of one is the same.Since coefficient of y is 1 in both equations,

α=m

β=a2m2b2

i.e.,β=a2α2b2

i.e.,β2=a2α2b2

a2α2b2=b2

since αβ is the locus points. this represents the curve,

i.e.,x2(ab)2y2b2=1 which is a hyperbola

Hence option (a) is correct.


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