Question

Find the equations of ellipses satisfying the following conditions :Foci at(3,0) and(-3,0) , passing through(4,1)

Answer 1

Dear student

The foci are on x-axis equidistant from origin. That means the axes of ellipse are coordinate axes and the centre of ellipse is (0,0)

The equation of such ellipse is

x² /a² + y² /b² = 1

a is semi major axis and b is semi minor axis

We know that its foci are given by (+-ae, 0)

e is eccentricity = √ 1 - b²/a²

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ae = 3

a² e² = 9

a² (1 - b² /a²) = 9

a² - b² = 9

a² = 9 + b²

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the ellipse passes through (4,1)

4² / a² + 1¹ / b² = 1

16/a² + 1/b² = 1

16 b² + a² = a² b²

16b² + 9 + b² = (9 + b²) b²

17 b² + 9 = 9b² + b⁴

b⁴ - 8b² - 9 = 0

(b² + 1)(b² - 9) = 0

b² - 9 = 0 is acceptable

b² = 9

a² = 9 + b² = 18

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Thus the equation of our ellipse is

x² /18 + y² /9 = 1

or

x² + 2y² = 18