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Question

The equation of directrix of a conic is
L:x+y1=0 and the focus is the point (0,0). Find the equation of the conic if its eccentricity is
12


A

3x2 + 3y2 + 2xy + 2x + 2y 1 = 0

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B

3x2 + 3y2 + 2xy + 2x + 2y + 1 = 0

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C

3x2 + 3y2 + 2xy 2x 2y + 1 = 0

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D

3x2 3y2 + 2xy + 2x 2y + 1 = 0

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Solution

The correct option is C

3x2 + 3y2 + 2xy 2x 2y + 1 = 0


we will use the definition of conic to arrive at its equation.

PSPM = e

P is our moving point and let its coordinate be (h,k)

PS = (h0)2 + (k0)2 = h2 + k2

PM = perpendicular distance of P from L

=h+k12

e = 12 (given)

PSPM = e h2 + k2 = 12 × h+k12

h2 + k2 = (h+k1)222

4h2 + 4k2 = h2 + k2 2hk + 1 2h 2k

3h2 + 3k2 2hk 2h 2k + 1 = 0
we will replace (h,k) with (x,y) to get the equation of conic
3x2 + 3y2 2xy 2x 2y + 1 = 0


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