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Question

Focus of hyperbola is (±3,0) and equation of tangent is 2x+y-4=0, find the equation of hyperbola is


A

4x25y2=20

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B

5x24y2=20

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C

4x25y2=1

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D

5x24y2=1

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Solution

The correct option is A

4x25y2=20


Explanation for correct option:

Step1. General equation hyperbola

General equation of hyperbola is given by

x2a2-y2b2=1...(i),also b2=a2(e2-1)

Step2. Finding value of a&b

Given, Foci of the hyperbola is (±3,0) i.e. ae=±3

b2=a2(e2-1)b2=(ae)2-a2b2=32-a2a2+b2=9...(ii)

Equation of tangent on hyperbola is 2x+y-4=0 touches equation (i) if a2×22-b2×12=42...(iii)

Now, Equation of tangent on curve is given by lx+my+n=0touches equation (i)if a2l2-b2m2=n2

Add equation (ii)&(iii)

a2+b2=94a2-b2=16-------5a2=25a2=5&b2=4

Step3. finding equation hyperbola.

Put the value of a2&b2 in equation (i)

Equation of parabola is

x25-y24=14x2-5y2=20

Hence correct option is (A)


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