Hyperbola finding equation?

Question

Endpoints of the transverse axis at (-8,2) and (0.2), and length of the latera recta = 6

Answer 1

NOTE :

The standard form of the equation of a hyperbola with center (where a and b are not equals to 0) is (Transverse axis is horizontal) or (Transverse axis is vertical).

Let the end points of the transverse axis are A(- 8, 2) and B(0, 2).

Length of the transverse axis is 2a.

2a = distance between A and B.

2a = √ [(- 8 - 0)^2 + (2 - 2)^2 ]

2a = √ [(- 8)^2 + (0)^2 ] = 8

 a = 4

⇒ a ^2 = 16.

Length of latus rectum = 2b ^2/a = 6

2b ^2/4 = 6

⇒ b ^2 = 12.

Mid point of the two extremes of transverse axis gives the center of hyperbola.
C(h, k )= [ (- 8 + 0)/2, (2 + 2)/2 ] = (- 4, 2).

Substitute the values of (h, k ) = (- 4, 2), a ^2 = 16, and b ^2 = 12 in standard form of hyperbola equation.

(x - (- 4))^2/16 - (y - 2)^2/12 = 1

(x + 4)^2/16 - (y - 2)^2/12 = 1

Therefore, the hyperbola equation is (x + 4)^2/16 - (y - 2)^2/12 = 1.