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 $(h,k)$ is $T=S_1$
$\Rightarrow \frac{xh}{25}-\frac{yk}{36}=\frac{h^2}{25}-\frac{k^2}{36}$
Since it passes through $\left(2,4\right)$
$\frac{2h}{25}-\frac{4k}{36}=\frac{h^2}{25}-\frac{k^2}{36}\Rightarrow \frac{h^2-2h+1}{25}-\frac{k^2-4k+4}{36}=\frac{1}{25}-\frac{4}{36}$
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