Question

Find the equation to the hyperbola, referred to its axes as axes of coordinates, whose transverse axis is $7$ and which passes through the point $(3,-2)$.

• A. $65y^2-16x^2=196$
• B. $65y^2-14x^2=196$
• C. $85y^2-16x^2=196$
• D. $85y^2-16x^2=147$

Answer 1

The correct option is C $85y^2-16x^2=196$

General equation of hyperbola is $\frac{y^2}{b^2}-\frac{x^2}{a^2}=1$

Length of transverse axis is $2a$.

So, $2a=7$

$\Rightarrow a=\frac{7}{2}$

Equation becomes,

$\frac{y^2}{b^2}-\frac{4x^2}{49}=1$

It passes through $(3,-2)$, so it should satisfy the parabola,

$\frac{4}{b^2}-\frac{36}{49}=1$

On solving, we get

$85y^2-16x^2=196$