The eccentricity of an ellipse, with its centre at the origin, is 1/2. If one of the directrices is x=4, then the equation of the ellipse is - Mathematics MCQ AIEEE 2004 2026 | ExamTest.live

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The eccentricity of an ellipse, with its centre at the origin, is \frac{1}{2} . If one of the directrices is x = 4, then the equation of the ellipse is

Solution & Step-by-step Explanation

For an ellipse with centre at the origin and directrix x = a / e : Given eccentricity e = 1/2 and directrix x = 4 : a / e = 4 \Rightarrow a / (1/2) = 4 \Rightarrow 2 a = 4 \Rightarrow a = 2 . The relationship between a, b, and e is b^{2} = a^{2} (1 - e^{2}) . b^{2} = 2^{2} (1 - (1/2)^{2}) = 4 (1 - 1/4) = 4 (3/4) = 3 . The equation of the ellipse is \frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1 : \frac{x ^{2}}{4} + \frac{y ^{2}}{3} = 1 \Rightarrow 3 x^{2} + 4 y^{2} = 12 .

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The eccentricity of an ellipse, with its centre at the origin, is \frac{1}{2} . If one of the directrices is x = 4, then the equation of the ellipse is

3 x^{2} + 4 y^{2} = 1

3 x^{2} + 4 y^{2} = 12

4 x^{2} + 3 y^{2} = 12

4 x^{2} + 3 y^{2} = 1

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