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Determine the equation of the ellipse with foci
(3,2) and
(-5,2), and a minor axis of length 6 .

Determine the equation of the ellipse with foci $(3,2)$ and $(-5,2)$ , and a minor axis of length $6$ .

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Write equations of ellipses in standard form using properties

Full solution

Q. Determine the equation of the ellipse with foci

(3,2)

and

(-5,2)

, and a minor axis of length

6

.

Find Center of Ellipse: We need to find the center of the ellipse, which is the midpoint between the foci. $\newline$ Center $(h, k) = (\frac{3 + (-5)}{2}, \frac{2 + 2}{2}) = (\frac{-2}{2}, \frac{4}{2}) = (-1, 2)$
Calculate Distance Between Foci: The distance between the foci is $2c$ , where $c$ is the distance from the center to a focus. $\newline$ Distance between foci = $|3 - (-5)| = 8$ $\newline$ So, $c = \frac{8}{2} = 4$
Determine Length of Minor Axis: The length of the minor axis is given as $6$ , which is $2b$ , where $b$ is the semi-minor axis. $\newline$ Therefore, $b = \frac{6}{2} = 3$
Find Value of Semi-Major Axis: We need to find the value of $a$ , the semi-major axis. The relationship between $a$ , $b$ , and $c$ for an ellipse is $c^2 = a^2 - b^2$ . We already know $b = 3$ and $c = 4$ , so we can solve for $a$ . $a^2 = c^2 + b^2 = 4^2 + 3^2 = 16 + 9 = 25$ Therefore, $a = \sqrt{25} = 5$
Write Equation in Standard Form: Now we can write the equation of the ellipse in standard form. $\newline$ The standard form of the equation of an ellipse with a horizontal major axis is: $\newline$ $\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1$ $\newline$ Substituting the values of $h$ , $k$ , $a$ , and $b$ , we get: $\newline$ $\frac{(x + 1)^2}{5^2} + \frac{(y - 2)^2}{3^2} = 1$
Simplify Equation: Simplify the equation to get the final standard form. $\newline$ $(x + 1)^2/25 + (y - 2)^2/9 = 1$

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