Question

HELP TIMER Write the equation of a hyperbola centered at the origin with x-intercept +/- 4 and foci of +/-2(squareroot 5)

Answer 1

Given:

x-intercepts of the hyperbola are ±4.

The foci of hyperbola are $\pm 2\sqrt{5}$.

Center of the hyperbola is at origin.

To find:

The equation of the hyperbola.

Solution:

The general equation of a hyperbola:

$\dfrac{(x-h)^2}{a^2}-\dfrac{(y-k)^2}{b^2}=1$            ...(i)

Where, (h,k) is the center of the hyperbola, ±a are x-intercepts, $(\pm c,0)$ are foci.

Center of the hyperbola is at origin. So, h=0 and k=0.

x-intercepts of the hyperbola are ±4. So,

$\pm a=\pm 4$

$a=4$

The foci of hyperbola are $\pm 2\sqrt{5}$.

$\pm c=\pm 2\sqrt{5}$

$c=2\sqrt{5}$

We know that,

$a^2+b^2=c^2$

$(4)^2+b^2=(2\sqrt{5})^2$

$16+b^2=20$

$b^2=20-16$

$b^2=4$

Taking square root on both sides, we get

$b=\sqrt{4}$                    [b>0]

$b=2$

Substituting $h=0,k=0,a=4,b=2$ in (i), we get

$\dfrac{(x-0)^2}{4^2}-\dfrac{(y-0)^2}{2^2}=1$

$\dfrac{x^2}{4^2}-\dfrac{y^2}{2^2}=1$

Therefore, the correct option is (d).