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Question

The equation of the ellipse whose focus is (1, -1). directrix x – y – 3 = 0 and eccentricity fraction numerator 1 over denominator 2 end fraction is

  1. 7x2 + 2xy + 7y2 – 10x + 10y + 7 = 0    
  2. 7x2 + 2xy + 7y2 = 7 = 0    
  3. 7x2 + 2xy + 7y2 + 10x – 10y – 7 = 0    
  4. none of these.    

Hint:

The ratio of distances from the center of the ellipse from either focus to the semi-major axis of the ellipse is defined as the eccentricity of the ellipse.
The eccentricity of ellipse, e = c/a
Where c is the focal length and a is length of the semi-major axis.
Since c ≤ a the eccentricity is always greater than 1 in the case of an ellipse.

The correct answer is: 7x2 + 2xy + 7y2 – 10x + 10y + 7 = 0



    We know that 
    The eccentricity of ellipse, e = c/a, where c is the focal length and a is length of the semi-major axis.
    e space equals space fraction numerator P S over denominator P M end fraction
rightwards double arrow P S space equals space e P M
rightwards double arrow open curly brackets left parenthesis x minus 1 right parenthesis squared space plus space left parenthesis y space plus 1 right parenthesis squared close curly brackets to the power of 1 half end exponent space equals space 1 half fraction numerator x minus y space minus space 3 over denominator square root of 2 end fraction
S q u a r i n g space o n space b o t h space s i d e s
rightwards double arrow left parenthesis 2 square root of 2 right parenthesis end root squared left parenthesis x squared space minus 2 x space plus space 1 space plus space y squared space plus 2 y space plus space 1 right parenthesis space equals space left parenthesis x squared space plus space y squared space plus space 9 space minus 2 x y space plus 6 y space minus 6 x

    rightwards double arrow 8 x squared space plus 8 y squared space minus 16 x space plus 16 y space plus 16 space minus x squared space minus y squared space plus 2 x y space minus 6 y space plus 6 x space minus space 9 space equals space 0
rightwards double arrow 7 x squared space plus 7 y squared space minus 10 x space plus 10 y space plus 7 space plus 2 x y equals space 0

    Thus, the equation of the ellipse whose focus is (1, -1). directrix x – y – 3 = 0 and eccentricity fraction numerator 1 over denominator 2 end fraction is 7 x squared space plus 7 y squared space minus 10 x space plus 10 y space plus 7 space plus 2 x y equals space 0

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