Question

The center of a hyperbola is located at the origin. One focus is located at (0, 20), and its associated directrix is represented by the line y = -256/20. What is the equation of the hyperbola?

Answer 1

Answer:

The equation of the hyperbola is $\frac{y^2}{256}-\frac{x^2}{144} =1$.

Step-by-step explanation:

Since directrix is $y=\frac{-256}{20}$, therefore the hyperbola is along the y-axis.

Focus of the hyperbola are (0,c) and (0,-c).

The given focus is (0,20), therefore the value of c is 20.

Directrix of the hyperbola is

$y=\pm \frac{a^2}{c}$

$\frac{-256}{20}=\pm \frac{a^2}{20}$

$256=a^2$

$a=16$

Therefore the value of a is 16.

The relation between a, b and c is

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

$b^2+(16)^2=(20)^2$

$b^2+256=400$

$b^2=400-256$

$b^2=144$

$b=12$

Therefore the value of b is 12.

The standard form of the hyperbola is,

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

Since the value of a is 16 and the value of b is 12, therefore

$\frac{y^2}{256}-\frac{x^2}{144} =1$

Answer 2

Answer:

D) $y^{2}/16^{2} - x^{2} /12^{2} =1$

Step-by-step explanation:

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