Question

The length of latus rectum of the hyperbola $xy-3x-3y+7=0$ is

• A. $4$
• B. $3$
• C. $2$
• D. $1$

Answer 1

Answer: (A)

$4$

Given Equation of Hyperbola is $xy-3y-3x+7=0$

We can rewrite the equation as $x(y-3)-3y+7=0$ .....$\left(1\right)$

Now by adding and subtracting $9$ in equation $\left(1\right)$, we get

$\Rightarrow$ $x(y-3)-3y+9-9+7=0$

Now eq. of Hyperbola can be written in the simple terms as

$x(y-3)-3(y-3)-2=0$

$\Rightarrow$ $(x-3)(y-3)=2$ or $(x-3)(y-3)=(\sqrt{2}{)}^2$ ......$\left(2\right)$

Equation $\left(2\right)$ is similar to equation of a rectangular Hyperbola of the form $xy=c^2$, with shifted origin at $(3,3)$

So given Hyperbola is also a rectangular Hyperbola, with $c=\sqrt{2}$

We know that for a rectangular Hyperbola $b=a=c\sqrt{2}$

So value of $a$ for given Hyperbola $=c\sqrt{2}=\sqrt{2}\times \sqrt{2}=2$

$\Rightarrow$ For any rectangular Hyperbola length of latusrectum $=2a$

Hence, the length of latusrectum of given Hyperbola $=2\times 2=4$