An ellipse is given by the equation 8x 2 + 2y 2 = 32 . Find a) the major axis an

Question

An ellipse is given by the equation 8x 2 + 2y 2 = 32 . Find a) the major axis and the minor axis of the ellipse and their lengths, b) the vertices of the ellipse, c) and the foci of this ellipse

Explanation / Answer

a) Divide all terms of the equation by 32 to obtain x 2 / 4 + y 2 / 16 = 1 The above equation may be written as follows x 2 / b 2 + y 2 / a 2 = 1 with a = 4 and b = 2 and a > b. Hence the major axis is the y axis and the minor axis is the x axis. The length of the major axis = 2a = 8 and the length of the minor axis = 2b = 4 b) The vertices are on the major axis at the points (0 , a) = (0 , 4) and (0 , -a) = (0 , -4) c) The foci are on the major axis at the points (0 , c) and (0 , -c) such that c 2 = a 2 - b 2 = 12. Hence the foci are at the points (0 , 2?3) and (0 , -2?3)

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