Find the equation of the hyperbola vertices

Question

(4,-1)(-2,-1) foci (5,-1)(-3,-1)?

Answer 1

The vertices are (4, - 1) and (- 2, - 1) foci (5, - 1)(- 3, - 1)

Since the y - coordinate is constant in the vertices and foci.

This is horizontal hyperbola.

Standard form of hyperbola

a = semi-transverse axis , b = semi-conjugate axis

center: (h, k ) Vertices: (h + a, k ), (h - a, k )

Foci: (h + c, k ), (h - c, k )

c is the distance from center to each focus.

 

Find the center (h, k)

vertices: (4, - 1) and (- 2, - 1)

So the y coordinate of the center of hyperbola is  - 1.

h + a = 4 ----> (1)

h - a = - 2 ------> (2)

Add the equations (1) & (2).

h + a + h - a = 4 - 2

2h = 2

h = 1

So x  coordinate of center is 1.

(h, k) = (1, -1)

 

Find the values of a, b.

Substitute the h value in equation (1).

1 + a = 4

a  =  3

foci: (5, - 1) and (- 3, - 1)

h + c = 5

Substitute the h value in above equation.

1 + c = 5

c = 4

c² = a² + b²

(4)² = (3)² + b²

16 - 9 = b²

b = √7

Substitute the a,b and center in standard form.

The hyperbola equation is (x  - 1)²/9 - (y + 1)²/7 = 1.