Does anyone know how to get this equation?

Question

Find the equation of the hyperbola with foci (0, ±3) and difference of the focal radii 4?

Answer 1

Let F ang G be the foci.

So: F = (0,-3) and G = (0,+3)

If P = (x,y) is a point on the hyperbola then:

PF = distance from F to P

PG = distance from G to P

And:

|PF - PG| = difference in focal radii = 4 ..... | |  is the absolute value

Now let us write out the distances involved (this might be more clear if you draw a diagram and carefully label everything):

PF = SQRT[x^2 + (y + 3)^2]

PG = SQRT[x^2 + (y - 3)^2]

So:

SQRT[x^2 + (y + 3)^2] - SQRT[x^2 + (y - 3)^2] = +/-4

The +/- because of the absolute value. This means that whatever is inside the absolute value signs can be either + or - and the equation will be true.

Can we simplify this:

SQRT[x^2 + (y + 3)^2] = SQRT[x^2 + (y + 3)^2] +/-4

[x^2 + (y + 3)^2] = [x^2 + (y - 3)^2] +/- 8SQRT[x^2 + (y - 3)^2] + 16

x^2 + y^2 + 6y + 9 = x^2 + y^2 - 6y + 9 +/- 8SQRT[x^2 + (y - 3)^2] + 16

12y - 16 = +/- 8SQRT[x^2 + (y - 3)^2]

3y - 4 = +/- 2SQRT[x^2 + (y - 3)^2]

9y^2 - 24y + 16 = 4[x^2 + y^2 - 6y + 9]

5y^2 - 4x^2 = 20

y^2/4 - x^2/5 = 1

Answer 2

Haven't got time to explain at the moment,

but I'm pretty sure it is : y^2/4 - x^2/5 = 1.

Hope that's correct - rush job.