Question

If the length of the transverse axis along X-axis with centre at origin of a hyperbola is $7$ and it passes through the point $(5,-2)$. The equation of the hyperbola is

• A. $\frac{4}{49}{x}^2-\frac{196}{51}{y}^2=1$
• B. $\frac{49}{4}{x}^2-\frac{51}{196}{y}^2=1$
• C. $\frac{4}{49}{x}^2-\frac{51}{196}{y}^2=1$
• D. None of these

Answer 1

Answer: (C)

$\frac{4}{49}{x}^2-\frac{51}{196}{y}^2=1$

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

Length of transverse axis $=2a=7$

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

To find vertex $(h,k),h=\frac{b}{4a}$

$\therefore h=\frac{1}{\sqrt{7^2+2}}\therefore b=4ah=4\times 7\times h$

$\therefore b=\frac{2\times 7}{\sqrt{7^2+2}}$

$a^2=\frac{49}{4}$ and $b^2=\frac{196}{51}$

$\Rightarrow \frac{x^2}{\frac{49}{4}}-\frac{y^2}{\frac{196}{51}}=1\Rightarrow \frac{4x^2}{49}-\frac{51y^2}{196}=1$