Question

Question 15 Find the equation of the hyperbola satisfying the give conditions: Foci (, passing through (2, 3)

Class X1 - Maths -Conic Sections Page 262

Answer 1

Foci( 0 , ±√10) and passing through (2, 3) .
Let standard equation of hyperbola is
$\frac{ {y}^{2} }{ {a}^{2} } - \frac{ {x}^{2} }{ {b}^{2} } = 1 -- - -(1)$

concept : if ( 0, ± c) are foci of above hyperbola
then, c² = a² + b² .

here foci ( 0, ±√10) so, c = √10
hence,. √10² = a² + b²
a² + b² = 10 -------------(2)

again, hyperbola passing through (2, 3) .
so, (2, 3) will satisfy equation (1)
$\frac{ {3}^{2} }{ {a}^{2} } - \frac{ {2}^{2} }{ {b}^{2} } = 1$

9b² - 4a² = a²b² ------------(3)

now, equation (2) and (3),
9( 10 - a²) - 4a² = (10 - a²)a²
90 - 9a² - 4a² = 10a² - a⁴
a⁴ - 23a² + 90 = 0
a⁴ - 18a² - 5a² + 90 = 0
a² = 18 , a² = 5

if a² = 18 , b² = 10 - 18 = -8 but b² can't be negative.
so, a² ≠ 18 hence, a² = 5
b² = 10 - a² from equation (2)
b² = 10 - 5 = 5

hence, equation of parabola is
$\frac{ {y}^{2} }{5} - \frac{ {x}^{2} }{5} = 1$

y² - x² = 5