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: