Question

What will be the equation of a line which is parallel to the asymptote $y=x$ of the rectangular hyperbola
$x^2-y^2=8$
and passes through the focus?

• A. $y=x+3$
• B. $y=x-2$
• C. $y=x-4$
• D. $y=x-1$

Answer 1

The correct option is C $y=x-4$

Given: Equation of asymptote $y=x$ and equation of rectangular hyperbola $x^2-y^2=8$
The standard rectangular hyperbola equation is $x^2-y^2=a^2$.

Comparing the standard rectangular hyperbola equation with the given rectangular hyperbola equation, we get $a^2=8$.

$\Rightarrow a=2\sqrt{2}$

The coordinates of foci of a rectangular hyperbola is given by $(\pm a\sqrt{2},0)$.
$\therefore$ The coordinates of focus will be $(\pm 2\sqrt{2}\times \sqrt{2},0)$.

$\Rightarrow (\pm 4,0)$

To Find: Equation of line

Since, the line runs parallel to the asymptote, the slope of the line will be equal to the slope of the asymptote.

From $y=x$, we have the slope $m=1$.

Since, we have the slope and a point, we can calculate the equation of line using the formula $(y-y_1)=m(x-x_1)$

We have, $x_1=4,y_1=0,m=1$

Substituting all the values in the straight line equation.

$(y-0)=1(x-4)$
$\Rightarrow y=x-4$
and
$(y-0)=1(x-(-4\left)\right)$
$\Rightarrow y=x+4$