Question

The shortest distance between the line $y=x$ and the hyperbola $\frac{x^2}{9}-\frac{y^2}{4}=1$ is

• A. $\frac{3}{2}$ unit
• B. $\frac{2}{3}$ unit
• C. $\sqrt{\frac{5}{2}}$ unit
• D. $9$ unit
  

Answer 1

The correct option is C $\sqrt{\frac{5}{2}}$ unit

Any tangent to the hyperbola is
$y=mx\pm \sqrt{9m^2-4}$
for line $y=x$ to have minimum distance from hyperbola tangent at the nearest point on the hyperbola should be parallel to given line
$\therefore m=1$
$\Rightarrow y=x\pm \sqrt{5}$
hence required distance
$=\frac{\sqrt{5}-0}{\sqrt{1+1}}=\sqrt{\frac{5}{2}}$ unit