Find the equation of the hyperbola whose foci are \((0, ±\sqrt{10})\) and passing through the point (2, 3).

Concept:

The hyperbola taking Y-axis as reference is of the form \(\frac{{{y^2}}}{{{a^2}}} - \frac{{{x^2}}}{{{b^2}}} = 1\) has:

• Centre is given by: (0, 0)
• Vertices are given by: (0, ± a )
• Foci are given by: (0, ± c)

Given:

Foci = \((0, ±√{10})\)

Point on the hyperbola is (2, 3).

Calculation:

Foci = \((0, ±√{10})\)

so, ae = √ 10

We know that b² = a² e² - a²

⇒ b2 = (√ 10)² - a2

⇒ b² = 10 - a²       .............(i)

substituting above in the equation of hyperbola,

⇒ \(\frac{{{y^2}}}{{{a^2}}} - \frac{{{x^2}}}{{{10-a^2}}} = 1\)

⇒ (2, 3) points satisfy the above equation

⇒ \(\frac{{{3^2}}}{{{a^2}}} - \frac{{{2^2}}}{{{10-a^2}}} = 1\)

⇒ a⁴ - 23a² + 90 = 0

⇒ (a² - 5)(b² - 18)=0

⇒ a² = 5, 18

From equation (i)

b² = 10 - 5 = 5

again b2 = 10 - 18 = - 8 (Not possible)

The required equation of hyperbola is 

⇒ \(\frac{{{y^2}}}{{{5}}} - \frac{{{x^2}}}{{{5}}} = 1\)

⇒ y2 - x2 = 5