Question

Find the equation of the hyperbola satisfying the given condition:

 foci (0, ± \[\sqrt{10}\], passing through (2, 3).

Answer 1

The foci of hyperbola are  \[\left( 0, \pm \sqrt{10} \right)\] that pass through  \[\left( 2, 3 \right)\].

Thus, the value of ae = \[\sqrt{10}. \]

By squaring both the sides, we get:

\[ \left( ae \right)^2 = 10\]

\[ \Rightarrow a^2 + b^2 = 10\]

\[ \Rightarrow b^2 = 10 - a^2\]

Let the equation of the hyperbola be 

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

It passes through 

\[\left( 2, 3 \right)\].

\[\Rightarrow \frac{3^2}{a^2} - \frac{2^2}{10 - a^2} = 1\]

\[ \Rightarrow 90 - 9 a^2 - 4 a^2 = 10 a^2 - a^4 \]

\[ \Rightarrow a^4 - 23 a^2 + 90 = 0\]

\[ \Rightarrow \left( a^2 - 18 \right)\left( a^2 - 5 \right) = 0\]

\[ \Rightarrow a^2 = 18, 5\]

Now, 

\[b^2 = - 8 \text { or } 5\]

If we neglect the negative value, then b² = 5.
Thus, the equation of the hyperbola is

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