Find the centre, eccentricity, foci and directions of the hyperbola
16x2 – 9y2 + 32x + 36y – 164 = 0
Given: 16x2 – 9y2 + 32x + 36y – 164 = 0
To find: center, eccentricity(e), coordinates of the foci f(m,n), equation of directrix.
16x2 – 9y2 + 32x + 36y – 164 = 0
⇒ 16x2 + 32x + 16 – 9y2 + 36y – 36 – 16 + 36 – 164 = 0
⇒ 16(x2 + 2x + 1) – 9(y2 – 4y + 4) – 16 + 36 – 164 = 0
⇒ 16(x2 + 2x + 1) – 9(y2 – 4y + 4) – 144 = 0
⇒ 16(x + 1)2 – 9(y – 2)2 = 144
Here, center of the hyperbola is (-1, 2)
Let x + 1 = X and y – 2 = Y
Formula used:
For hyperbola
Eccentricity(e) is given by,
Foci are given by (±ae, 0)
The equation of directrix are
Length of latus rectum is
Here, a = 3 and b = 4
Therefore,
⇒ X = ±5 and Y = 0
⇒ x + 1 = ±5 and y – 2 = 0
⇒ x = ±5 – 1 and y = 2
So, Foci: (±5 – 1, 2)
Equation of directrix are: