Given: Foci (0 \(\pm \sqrt{10}\)) passing through (2, 3)
To find: equation of the hyperbola
Formula used:
The standard form of the equation of the hyperbola is,
\(\frac{x^2}{a^2} - \frac{y^2}{b^2}\) = -1
Coordinates of the foci for a standard hyperbola is given by (0, ±be)
According to the question:
be = \(\sqrt{10}\)
⇒ b2e2 = 10
Since (2, 3) passing through hyperbola
\(\frac{x^2}{a^2} - \frac{y^2}{b^2}\) = -1
Therefore,
{∵ a2 = b2(e2 – 1)}
⇒ 90 – 13b2 = (10 – b2)b2
⇒ 90 – 13b2 = 10b2 – b4
⇒ 90 – 13b2 – 10b2 + b4 = 0
⇒ b4 – 23b2 + 90 = 0
⇒ b4 – 18b2 – 5b2 + 90 = 0
⇒ b2(b2 – 18) – 5(b2 – 18) = 0
⇒ (b2 – 18)(b2 – 5) = 0
⇒ b2 = 18 or 5
Case 1:
b2 = 18 and b2e2 = 10
a2 = b2(e2 – 1)
⇒ a2 = b2e2 – b2
⇒ a2 = 10 – 18
⇒ a2 = – 8
Hence, equation of hyperbola is:
Case 2:
b2 = 5 and b2e2 = 10
a2 = b2(e2 – 1)
⇒ a2 = b2e2 – b2
⇒ a2 = 10 – 5
⇒ a2 = 5
Hence, equation of hyperbola is: