Question

The combined equation of the asymptotes for the hyperbola $x^2-2y^2=2$

• A.
• B.
• C.
• D.

Answer 1

The correct option is C

Given hyperbola is $x^2-2y^2=2$

$\frac{x^2}{2}-\frac{y^2}{1}=1$

Comparing this equation with standard hyperpola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$

$a=\sqrt{2},b=1$

Equation of pair of asymptotes $\frac{x^2}{2}-\frac{y^2}{1}=0$

$y^2=\frac{x^2}{2}$

Let's solve this problem without using the formula.

Definition of asymptotes:If the length of the perpendicular let fall from a point on ahyperbola to a straight line tends to zero as the point on the hyperbola moves to infinity along the hyperbola, the straight line is called asymptotes of the hyperbola.

Let y=mx+c is an asymptote of the hyperbola. Substituting the y=mx+c in the hyperbola

$x^2-2(mx+c{)}^2=2$

$x^2-2m^2{x}^2-2c^2-4mcx-2=0$

$x^2(1-2m^2)-4mcx-2-2c^2=0$ .........(1)

if the line y=mx+c is an asymptote to the given hyperbola then it touches the hyperbola at infinity

So, both the roots of the equation (1) must be infinity.