Question

Which of the following statements are correct?

1. The particular kind of hyperbola in which the lengths of the transverse and conjugate axis are equal is called an equilateral hyperbola.

2. Eccentricity of equilateral hyperbola = $\sqrt{2}$
3. Equation of pair of asymptotes of rectangular hyperbola $x^2-y^2=a^2$ is $x^2-y^2=0$

4. Equation of pair of asymptotes of rectangular hyperbola $x^2-y^2=a^2$ is $x^2-y^2=-a^2$

• A. only 1 & 2
• B. only 1, 2 & 3
• C. only 2, 3 & 4
• D. All of them

Answer 1

The correct option is B

only 1, 2 & 3

Definition of rectangular: It is a type of hyperbola in which the lengths of the transverse and conjugate axis are equal.

$2a=2b$

$a=b$

Equation of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$

Here,$a=b$

Equation of the rectangular hyperbola

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

$x^2-y^2=a^2$

Eccentricity of equilateral hyperbola $e=\sqrt{1+\frac{b^2}{a^2}}$

$a=b$

$e=\sqrt{1+1}=\sqrt{2}$

Equation of the asymptotes $y=\pm \frac{b}{a}x$

$y=\pm 1.x$

$y=\pm x$

$y=x$ & $y=-x$

Equation of pair of asymptotes $(y-x)(y+x)=0$

$y^2-x^2=0or{x}^2-y^2=0$

so,only statements 1,2 & 3 are correct