Let the eccentricity of the hyperbola x 2 a 2 - y 2 b 2 = 1 be reciprocal to that of the ellipse x 2 + 4 y 2 = 4 . If the hyperbola passes through a focus of the ellipse, then [2011]

Question

Let the eccentricity of the hyperbola $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$ be reciprocal to that of the ellipse $x^{2} + 4 y^{2} = 4 .$ If the hyperbola passes through a focus of the ellipse, then [2011]

Smart Achievers · Accepted Answer

(2, 4)

Given ellipse $x^{2} + 4 y^{2} = 4$ $\Rightarrow \frac{x^{2}}{4} + \frac{y^{2}}{1} = 1$

Its focus is $(\pm \sqrt{3}, 0)$ and eccentricity, $e = \sqrt{1 - \frac{1}{4}} = \frac{\sqrt{3}}{2}$

Given hyperbola $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$

Its eccentricity is $\sqrt{1 + \frac{b^{2}}{a^{2}}}$

According to the question, $\sqrt{1 + \frac{b^{2}}{a^{2}}} = \frac{2}{\sqrt{3}}$ $\Rightarrow \frac{b}{a} = \frac{1}{\sqrt{3}}$

As hyperbola passes through the eccentricity of the ellipse $(\pm \sqrt{3}, 0)$

$∴$ $\frac{3}{a^{2}} = 1$ or $a = \sqrt{3}$ $∴$ $b = 1$ and focus of hyperbola $(\pm 2, 0)$

$∴$ Equation of hyperbola is $\frac{x^{2}}{3} - \frac{y^{2}}{1} = 1$ $\Rightarrow x^{2} - 3 y^{2} = 3$

Solution

Q.
Let the eccentricity of the hyperbola $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$ be reciprocal to that of the ellipse $x^{2} + 4 y^{2} = 4 .$ If the hyperbola passes through a focus of the ellipse, then [2011]

1 the equation of the hyperbola is $\frac{x^{2}}{3} - \frac{y^{2}}{2} = 1$

2 a focus of the hyperbola is $(2, 0)$

3 the eccentricity of the hyperbola is $\sqrt{\frac{5}{3}}$

4 the equation of the hyperbola is $x^{2} - 3 y^{2} = 3$

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Solution

Q. Let the eccentricity of the hyperbola x2a2-y2b2=1 be reciprocal to that of the ellipse x2+4y2=4. If the hyperbola passes through a focus of the ellipse, then [2011]

1 the equation of the hyperbola is x23-y22=1

2 a focus of the hyperbola is (2,0)

3 the eccentricity of the hyperbola is 53