Question

Minimum length of a normal chord to the hyperbola $xy=c^2$ lying between different branches is

• A. $2\sqrt{2}$
• B. $2\sqrt{2}c$
• C. $\sqrt{2}c$
• D. None of these

Answer 1

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

For a given rectangular hyperbola $xy=c^2$, the two vertices are $A(a,0)$ and $B(-a,0)$, one each side of the hyperbola.

Both the vertices lie on the transverse axis of the hyperbola and we know that transverse axis of the hyperbola is always perpendicular or normal to the hyperbola.

We can see from the figure that minimum length of the normal chord to the hyperbola lying between different branches of the given hyperbola is the length of the transverse axis or $AB$.

$\Rightarrow$ $AB=2a$

For a rectangular hyperbola $a=c\sqrt{2}$

So $AB=2a=2\times c\sqrt{2}=2\sqrt{2}c$

Hence the minimum length of the normal chord to the given hyperbola $xy=c^2$ lying between different branches is $AB=2\sqrt{2}c$.

Correct option is $B$.