The equation of an ellipse whose eccentricity | vedclass

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Question

(A) The length of the major axis is the distance between the vertices $(4, 0)$ and $(10, 0)$,which is $10 - 4 = 6$.Thus,$2a = 6$,so $a = 3$.The center

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$3x^2 + 4y^2 - 42x + 120 = 0$

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(A) The length of the major axis is the distance between the vertices $(4, 0)$ and $(10, 0)$,which is $10 - 4 = 6$.Thus,$2a = 6$,so $a = 3$.The center of the ellipse is the midpoint of the vertices: $(\frac{4+10}{2}, 0) = (7, 0)$.Given eccentricity $e = 1/2$,we find $b^2 = a^2(1 - e^2) = 3^2(1 - (1/2)^2) = 9(3/4) = 27/4$.The equation of the ellipse is $\frac{(x - 7)^2}{a^2} + \frac{(y - 0)^2}{b^2} = 1$,which is $\frac{(x - 7)^2}{9} + \frac{y^2}{27/4} = 1$.Multiplying by $27$,we get $3(x - 7)^2 + 4y^2 = 27$.$3(x^2 - 14x + 49) + 4y^2 = 27$.$3x^2 - 42x + 147 + 4y^2 = 27$.$3x^2 + 4y^2 - 42x + 120 = 0$.

The equation of an ellipse whose eccentricity is $1/2$ and the vertices are $(4, 0)$ and $(10, 0)$ is

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