Question

The transverse axis of a hyperbola is of length 2a and a vertex divides the segment of the axis between the centre and the corresponding focus in the ratio 2 : 1, then the equation of the hyperbola is

• A. $4x^2-5y^2=4a^2$
• B. $4x^2-5y^2=5a^2$
• C. $5x^2-4y^2=4a^2$
• D. $5x^2-4y^2=5a^2$.

Answer 1

The correct option is D $5x^2-4y^2=5a^2$.

Clearly, $\frac{2ae}{3}=a\Rightarrow e=\frac{3}{2}$

So, $S=(\frac{3a}{2},0)$

Directrix is $x=\frac{a}{e}=\frac{a}{\frac{3}{2}}=\frac{2a}{3}$

So, eq of hyperbola will be ${(x-\frac{3a}{2})}^2+y^2=\frac{9}{4}{(x-\frac{2a}{3})}^2$

$x^2+\frac{9a^2}{4}-3ax+y^2=\frac{9}{4}(x^2+\frac{4a^2}{9}-\frac{4ax}{3})$

$x^2+\frac{9a^2}{4}-3ax+y^2=\frac{9x^2}{4}+a^2-3ax$

$4x^2+9a^2+4y^2=9x^2+4a^2$

$5x^2-4y^2=5a^2$.