Question

Length of the latus rectum of the hyperbola $xy=c^2$, is

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

Answer 1

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

$Given$
$equation\text{of rectangular hyperbola is}$
$\text{xy=}{\text{c}}^2............\left(i\right)$
$We\text{know that in a rectangular hyperbola the length of transverse axis, length of conjugate axis}$
$\text{and length of latus rectum all are equal and it is along}$
$\text{y=x}...........\left(ii\right)$
$so\text{from}\left(i\right)$
$x\cdot x=c^2$
$takingsquarerootbothsides$
$x=\pm c$
$Hencefrom\left(ii\right)$
$y=\pm c$
$Hence(x,y)=(c,c,)and(-c.-c)$
$We\text{know}\text{distance between two points}(x_1,y_1)and(x_2,y_2)is$
$d=\sqrt{{(x_2-x_1)}^2+{(y_2-y_1)}^2}$
$Hencedis\tan cebetween(c,c,)and(-c.-c)$
$d=\sqrt{{(-c-c)}^2+{(-c-c)}^2}$
$=\sqrt{4c^2+4c^2}$
$=2\sqrt{2}c$