Find equation for hyperbola?

Question

Center (2,2), Vertex (2,4),

Answer 1

The standard form of the equation of a hyperbola with center (h, k) (where a and b are not equals to 0) is (x - h)²/a² - (y - k)²/b² = 1 (Transverse axis is Horizontal) or (y - k)²/a² - (x - h)²/b² = 1 (Transverse axis is Vertical).

The vertices and foci are, respectively a and c units from the center (h, k) and the relation between a, b and c is b² = c² - a².The center of the hyperbola lies at the midpoint of its vertices or foci.

The center of the hyperbola is (2, 2) and its vertex is (2, 4).

Since the x - coordinate is same in the center and vertex, this is a Vertical hyperbola.

Vertical hyperbola : (y - k)²/a² - (x - h)²/b² = 1

Center = (h, k) = (2, 2) ------> (y - 2)²/a² - (x - 2)²/b² = 1.

Here find the value of a and b.

Vertices = (h, k ± a) = (2, 4)

k ± a = 4

2 + a = 4 and 2 - a = 4

a = 2 and a = - 2.

The equation of hyperbola is (y - 2)²/4 - (x - 2)²/b² = 1.

There is no possibility to find the value of b by using given data.

To find the equation of hyperbola, mention the any information for the value of b.

Equation form	(y - k)2/a2 - (x - h)2/b2 = 1
Center	(h, k)
Vertices	(h, k ± a)
Co - vertices	(h ± b, k)
Transverse axis	Vertical
Transverse length	2a
Conjugate axis	Horizontal
Conjugate length	2b
Foci	(h, k ± c)
Asymptotes	y =  ± a/b (x - h) + k

The equation of hyperbola is (y - 2)²/4 - (x - 2)²/b² = 1.