Question

Let F(0,5) and ????(0, −5) be the foci of a hyperbola. Let the points P(x, y) on the hyperbola satisfy either
PF − PG = 6 or PG− PF = 6. Use the distance formula to derive an equation for this hyperbola, writing your
answer in the form
x2 / a2 − y2 / b2 = 1.

Answer 1

Answer:

Step-by-step explanation:

Remember, the points P on the hyperbola satisfy that the value absolute of the difference of the distances of P to the foci is constant and less than the distance between the foci.

Then

$\lvert\lvert PF\lvert\lvert-\lvert\lvertPG\lvert\lvert=2a, \; \lvert\lvert PG\lvert\lvert-\lvert\lvert PF\lvert\lvert=2a$

Therefore, $2a=6\\a=3$

Also, the foci $(0,c)=(0,5), \; (0,-c)=(0,5)$ satisfy that $c=\sqrt{a^2+b^2}$, then

$5=\sqrt{3^2+b^2}\\5^2=3^2+b^2\\25-9=b^2\\16=b^2$

Then, the equaton of the hyperbola is

$\frac{x^2}{3^2}-\frac{y^2}{16}=1\\\frac{x^2}{9}-\frac{y^2}{16}=1$