Question

Find the equation of the hyperbola satisfying the give conditions: Foci (0, \(±\sqrt{10}\)), passing through (2, 3)

Answer 1

Foci (0, \(±\sqrt{10}\)) and passing through (2, 3)
Here, the foci are on the y-axis. 
Therefore, the equation of the hyperbola is of the form \(\frac{y^2}{a^2} –\frac{ x^2}{b^2} = 1\)

Since the foci are\( (±\sqrt{10}, 0), c = \sqrt{10}\)

We know that \(a^2 + b^2 = c^2\)
\(∴ b^2 = 10 – a^2 ………….. (1)\)

Since the hyperbola passes through point (2, 3), 

\(\frac{9}{a^2} – \frac{4}{b^2} = 1 … (2)\)
From equations (1) and (2), we obtain

\(\frac{9}{a^2} – \frac{4}{(10-a^2)} = 1\)

\(⇒ 9(10 – a^2) – 4a^2 = a^2(10 –a^2)\)

\(⇒ 90 – 9a^2 – 4a^2 = 10a^2 – a^4\)

\(⇒ a^4 – 23a^2 + 90 = 0\)

\(⇒ a^4 – 18a^2 – 5a^2 + 90 = 0\)

\(⇒ a^2(a^2 -18) -5(a^2 -18) = 0\)

\(⇒ (a^2 – 18) (a^2 -5) = 0\)

\(⇒ a^2 = 18\) or \(5\)

In hyperbola, \(c > a, i.e., c^2 > a^2\)

\(∴ a^2 = 5\)
\(⇒ b^2 = 10 – a^2= 10 – 5= 5\)

Thus, the equation of the hyperbola is \(\frac{y^2}{5} – \frac{x^2}{5} = 1\)