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**Illustration 1: If e _{1} is the eccentricity of the ellipse x^{2}/16 + y^{2}/25 = 1 and e_{2} is the eccentricity of the hyperbola passing through the foci of the ellipse and e_{1}e_{2} = 1, then find the equation of the hyperbola.(2006)**

e

e

This is obtained using the relation e

Hence, the foci of the ellipse are (0, ± 3)

Hence, the equation of the hyperbola is x

**Illustration 2: A hyperbola, having the transverse axis of length 2 sin θ, is confocal with the ellipse 3x ^{2} + 4y^{2} = 12. Then its equation is (2007)**

This gives a = 2 and b = √3

Hence, 3 = 4(1-e

So, ae = 2.1/2 = 1

Hence, the eccentricity e

1 = e

So, b

Hence, the equation of hyperbola is x

Or x

**Illustration 3: The circle x ^{2} + y^{2} – 8x = 0 and the hyperbola x^{2}/9 - y^{2}/4 = 1 intersect at the points A and B. (2010)**

Equation of tangent to circle is y = m(x-4) + √16m

These two equations will be identical for m = 2/√5

Hence, the equation of common tangent is 2x - √5y + 4 = 0

(2) The equation of the hyperbola is x

For their points of intersection x

So, this gives 4x

So, 13x

This gives x = 6 and -13/6

But x = -13/6 is not acceptable

Now, x = 6, y = ± 2√3

Required equation is (x-6)

This gives x

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