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Question

What is the condition for a line y=mx+c to be tangent to the hyperbola x2a2y2b2=1.


A

c=±b2a2m2

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B

c=±a2m2b2

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C

c=±a2b2m2

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D

c=±b2+a2m2

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Solution

The correct option is A

c=±b2a2m2


A line y = mx + c becomes a tangent to a curve when on solving the equations of the line and curve we get one solution.

Here solving these equation gives,

x2a2(mx+c)2b2=1

b2x2a2(mx+c)2=a2b2

x2.(b2a2m2)2a2mcxa2c2a2b2=0

We get one solution if Δ=0.

i.e.,4a4m2c2+4(b2a2m2)a2(c2b2)=0

a2m2c2+(b2c264a2m2c2+a2b2m2)=0

c2b2+a2m2=0

c2=b2a2m2.

c=±b2a2m2


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