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The equation of the conic section whose focus is at (-1,0) , directrix is the line 4x-3y+2=0 and eccentricity is (1/2)^1/2 is

 

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1 Answers

Latika Leekha
askIITians Faculty 165 Points
6 years ago
Hello student,
The eccentricity is given to be √½ < 1. Hence, the given conic section is ellipse.
Now, the focus of the ellipse is F(-1, 0) and directrix is the line 4x-3y + 2 = 0.
Let P(x, y) be any point on ellipse and |MP| be the perpendicular distance from P to the driectrix.
Then, |FP| = e |MP|
So,
\sqrt{\left (x+1)^{2}+y^{2} \right } = \sqrt{\frac{1}{2}}\frac{|4x-3y+2|}{\sqrt{4^{2}+3^{2}}}
This gives 50 (x2 + 1 + 2x + y2) = 16x2 + 9y2 + 4 – 24xy -12y +16x
On simplification, we get
34x2 + 41y2 + 84x + 12y + 24xy + 46 = 0.

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