# Write the Equation of the Hyperbola of Eccentricity √ 2 , If It is Known that the Distance Between Its Foci is 16. - Mathematics

Write the equation of the hyperbola of eccentricity $\sqrt{2}$,  if it is known that the distance between its foci is 16.

#### Solution

The foci of the hyperbola are of the form  $\left( ae, 0 \right)$ and  $\left( - ae, 0 \right)$.

Distance between the foci = $\sqrt{\left( ae - ( - ae \right)^2 + 0^{{}^2}}$

$= \sqrt{\left( 2ae \right)^{{}^2}}$

$= 2ae$

Distance between the foci is 16 and eccentricity of the hyperbola is  $\sqrt{2}$.

$\therefore 2ae = 16$

$\Rightarrow 2\sqrt{2}a = 16$

$\Rightarrow a = 4\sqrt{2}$

Now, $b^2 = a^2 ( e^2 - 1)$

$\Rightarrow b^2 = \left( 4\sqrt{2} \right)^2 ((\sqrt{2} )^2 - 1)$

$\Rightarrow b^2 = 32$

Equation of the hyperbola is given below:

$\frac{x^2}{\left( 4\sqrt{2} \right)^2} - \frac{y^2}{32} = 1$

$\Rightarrow \frac{x^2}{32} - \frac{y^2}{32} = 1$

Is there an error in this question or solution?

#### APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 27 Hyperbola
Q 4 | Page 18