​ In each of the following find the equations of the hyperbola satisfying the given conditions foci(0 ±√10) , passing through (2, 3) ​

Question

In each of the following find the equations of the hyperbola satisfying the given conditions foci(0 \(\pm\)√10) , passing through (2, 3)

Answer 1

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: