Question

Find the equation of ellipse whose focus is (-1 1) whose directrix is the line x-y+3=0 and whose eccentricity is 1/2

Answer 1

equation of ellipse is 7x² + 7y² + 2xy - 10x + 10y + 7 = 0.

from the definition of ellipse, ellipse is the locus of a point P(x, y) whose distance from the focus is product of eccentricity and perpendicular distance from point P and foot of perpendicular from the point to the directrix.

if S is focus , M is foot of perpendicular from the point P to the directrix and e is eccentricity.

then, SP = ePM

here, S = (-1, 1),so SP = √{(x + 1)² + (y - 1)²}

PM = |x - y + 3|/√2 and e = 1/2

so, √{(x + 1)² + (y - 1)²} = 1/2|x - y + 3|/√2

squaring both sides,

(x + 1)² + (y - 1)² = 1/8(x - y + 3)²

⇒7x² + 7y² + 2xy - 10x + 10y + 7 = 0

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