Question

The locus of the midpoint of the chords of the hyperbola $\frac{x^{2}}{25}-\frac{y^{2}}{36}=1$ which passes through the point $\left(2 , 4\right)$ is a hyperbola, whose transverse axis length (in units) is equal to

• A. $\frac{16}{5}$
• B. $\frac{4}{3}$
• C. $\frac{8}{5}$
• D. $\frac{61}{25}$

Answer 1

Answer: (A)

The equation of the chord whose mid-point is $\left(\right.h,k\left.\right)$ is $T=S_{1}$
$\Rightarrow \frac{x h}{25}-\frac{y k}{36}=\frac{h^{2}}{25}-\frac{k^{2}}{36}$
Since it passes through $\left(2 , 4\right)$
$\frac{2 h}{25}-\frac{4 k}{36}=\frac{h^{2}}{25}-\frac{k^{2}}{36}\Rightarrow \frac{h^{2} - 2 h + 1}{25}-\frac{k^{2} - 4 k + 4}{36}=\frac{1}{25}-\frac{4}{36}$
Hence, the locus of $\left(h , k\right)$ is
$\frac{\left(x - 1\right)^{2}}{\frac{16}{9}}-\frac{\left(y - 2\right)^{2}}{\frac{64}{25}}=-1$
This is the equation of a hyperbola whose transverse axis length is $\frac{16}{5}$ units