Question

A hyperbola passes through the points $(2,\sqrt{3})$ and $(4,\sqrt{21})$. If its foci are along x-axis and its axes are along coordinate axes then the length of its transverse axis is

• A. $4$
• B. $2\sqrt{2}$
• C. $2\sqrt{3}$
• D. $6$

Answer 1

The correct option is B $2\sqrt{2}$

If foci are along $x$-axis and axes of the hyperbola are along the coordinate axes then hyperbola is a standard form of the hyperbola. i.e. $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$

The hyperbola passes through two given points $(2,\sqrt{3})$ and $(4,\sqrt{21})$

Putting these points into equation of the hyperbola we get,

$\frac{2^2}{a^2}-\frac{(\sqrt{3}{)}^2}{b^2}=1$ or $\frac{4}{a^2}-\frac{3}{b^2}=1$ ...$\left(1\right)$

$\frac{4^2}{a^2}-\frac{(\sqrt{21}{)}^2}{b^2}=1$ or $\frac{16}{a^2}-\frac{21}{b^2}=1$ ....$\left(2\right)$

Multiplying eq$\left(1\right)$ with $7$ and subtracting eq.$\left(2\right)$ from it, we get,

$\to \frac{12}{a^2}=6$

$\to a=\sqrt{2}$

Hence length of it's transverse axis $=2a=2\sqrt{2}$

Correct option is $B$