If the eccentricities of the hyperbola x2a2 - y2b2 - 1 and y2b2 - x2a2 - 1 be e and e1- then 1e2 - 1e12 -

The correct option is B 1 Eccentricity of hyperbola is always gt; 1Hence 1e lt; 1 #160; so #160; 1e+1e_1 lt;2 Eccentricity of x ...

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Standard XII

Mathematics

Eccentricity

If the eccent...

Question

If the eccentricities of the hyperbola $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$ and $\frac{y^{2}}{b^{2}} - \frac{x^{2}}{a^{2}} = 1$ be $e$ and $e_{1}$ , then $\frac{1}{e^{2}} + \frac{1}{e_{1}^{2}} =$ .

1

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Solution

The correct option is B $1$
Eccentricity of hyperbola is always > 1
Hence $\frac{1}{e} < 1$ so $\frac{1}{e} + \frac{1}{e_{1}} < 2$

Eccentricity of $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$ is $e = \sqrt{1 + \frac{b^{2}}{a^{2}}}$
In case of the second hyperbola the axes gets interchanged and eccentricity is defined as the deviation of the curve from being circular.
$⟹$ $e_{1} = \sqrt{1 + \frac{a^{2}}{b^{2}}}$

$⟹$ $\frac{1}{e^{2}} + \frac{1}{e_{1}^{2}} =$ $\frac{a^{2}}{a^{2} + b^{2}} + \frac{b^{2}}{a^{2} + b^{2}} = 1$

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Similar questions

If the eccentricities of the hyperbolas $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1 a n d \frac{y^{2}}{b^{2}} - \frac{x^{2}}{a^{2}} = 1$ be e and $e_{1}$ , then $\frac{1}{e^{2}} + \frac{1}{e_{1}^{2}} =$

Q. If eccentricities of two hyperbolas

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

and

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

are

e_{1}

and

e_{2}

respectively then prove that

\frac{1}{e_{1}^{2}} + \frac{1}{e_{2}^{2}} = 1

Q. If the eccentricity of the hyperbola

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

e

then the eccentricity of the hyperbola

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

is :

If $e_{1} a n d e_{2}$ are the eccentricity of hyperbola $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$ and $\frac{y^{2}}{a^{2}} - \frac{x^{2}}{b^{2}} = 1$ , then point $\frac{1}{e_{1}}, \frac{1}{e_{2}}$ lies on the circle.

Q. The ellipse

\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1

and hyperbola

\frac{x^{2}}{A^{2}} - \frac{y^{2}}{B^{2}} = 1

are having a same foci and length of minor axis of ellipse is same as the conjugate axis of the hyperbola. If

e_{1}

e_{2}

are the eccentricities of ellipse and hyperbola respectively, then the value of

\frac{1}{e_{1}^{2}} + \frac{1}{e_{2}^{2}}

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If the eccentricities of the hyperbola x2a2−y2b2=1 and y2b2−x2a2=1 be e and e1, then 1e2+1e21=.

If the eccentricities of the hyperbola $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$ and $\frac{y^{2}}{b^{2}} - \frac{x^{2}}{a^{2}} = 1$ be $e$ and $e_{1}$ , then $\frac{1}{e^{2}} + \frac{1}{e_{1}^{2}} =$ .