(i) Centre (–1, 2), e = , foci = (4, 2), (–6, 2), x = – and x =

(ii) 7x 2 – 2y 2 + 12xy – 2x + 14y – 22 = 0

Sol. (i) Here 16x 2 + 32x + 16 – (9y 2 – 36y + 36) – 144 = 0 or 16(x + 1) 2 – 9(y – 2) 2 = 144

∴ = 1

Putting x + 1 = X and y – 2 = Y, the equation becomes = 1 which is in the standard form.

Here a 2 = 9 and b 2 = 16

 b 2 = a 2 (e 2 – 1), we get 16 = 9(e 2 – 1)

∴ e 2 – 1 =

∴ e 2 = , i.e. e =

Now, centre = (0, 0) X, Y = (–1, 2)

{  when X = 0, x + 1 = X gives x = –1 and when Y = 0, y – 2 = Y gives y = 2}

foci = (±ae, 0) X, Y = = (± 5, 0) X, Y = (– 1 ± 5, 2) = (4, 2), (–6, 2)

Directrices in X, Y coordinates have the equations

X ± = 0 or x + 1 ± = 0 i.e. x + 1 ± = 0

∴ x = – and x =

(ii) by PS = ePM (PS = ePM ls )

= ⇒ 7x 2 – 2y 2 + 12xy – 2x + 14y – 22 = 0.

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