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15. Find the equations of the hyperbola satisfying the given conditions.

Foci $(0,\pm\sqrt{10})$ , passing through (2,3)

Answers (1)

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Given, in a hyperbola,

Foci $(0,\pm\sqrt{10})$ , passing through (2,3)

Since foci of the hyperbola are in Y-axis, the equation of the hyperbola will be of the form ;

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

By comparing standard parameter (foci) with the given one, we get

$c=\sqrt{10}$

Now As we know, in a hyperbola

$a^2+b^2=c^2$

$a^2+b^2=10\:\:\:\:\:\:\:....(1)$

Now As the hyperbola passes through the point (2,3)

$\frac{3^2}{a^2}-\frac{2^2}{b^2}=1$

$9b^2-4a^2=a^2b^2\:\;\;\:\:\;\:....(2)$

Solving Equation (1) and (2)

$9(10-a^2)-4a^2=a^2(10-a^2)$

$a^4-23a^2+90=0$

$(a^2)^2-18a^2-5a^2+90=0$

$(a^2-18)(a^2-5)=0$

$a^2=18\:or\:5$

Now, as we know that in a hyperbola $c$ is always greater than, $a$ we choose the value

$a^2=5$

$b^2=10-a^2=10-5=5$

Hence The Equation of the hyperbola is

$\frac{y^2}{5}-\frac{x^2}{5}=1$

Posted by

Pankaj Sanodiya

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