# What is equation of hyperbola given Transverse axis on the y-axis, (2, √2) and (0, -1) are points on the curve?

Aug 3, 2018

${x}^{2} / 4 - {y}^{2} = - 1 , \mathmr{and} , 4 {y}^{2} - {x}^{2} = 4$.

#### Explanation:

We are given that the Transverse Axis of the required

Hyperbola is on the $Y - \text{Axis}$.

So, we suppose that its equation is $S : {x}^{2} / {a}^{2} - {y}^{2} / {b}^{2} = - 1$.

$\left(2 , \sqrt{2}\right) \in S . \therefore {2}^{2} / {a}^{2} - {\left(\sqrt{2}\right)}^{2} / {b}^{2} = - 1 , \mathmr{and} ,$

$\frac{4}{a} ^ 2 - \frac{2}{b} ^ 2 = - 1. \ldots \ldots \ldots \ldots . . \left({\ast}_{1}\right)$.

Again, $\left(0 , - 1\right) \in S \Rightarrow - \frac{1}{b} ^ 2 = - 1 , i . e . , {b}^{2} = 1. \ldots \ldots \ldots \ldots \ldots . \left({\ast}_{2}\right)$.

(ast_2) & (ast_1) rArr 4/a^2=2/b^2-1=1rArr a^2=4...(ast_3).

$\left({\ast}_{2}\right) \mathmr{and} \left({\ast}_{3}\right)$ give the desired equation of the hyperbola as

$S : {x}^{2} / 4 - {y}^{2} = - 1 , \mathmr{and} , 4 {y}^{2} - {x}^{2} = 4$.