Q19 of 27 Page 11

Find the centre, eccentricity, foci and directions of the hyperbola 16x2 – 9y2 + 32x + 36y – 164 = 0.

Consider 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



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:





⇒5X = ±9


⇒5X ± 9 = 0


⇒5(x+1) ± 9 = 0


⇒5x + 5 ± 9=0


⇒5x + 5 – 9 = 0 and 5x + 5 + 9 = 0


⇒5x – 4 = 0 and 5x + 14 = 0


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