Determine the standard equation of the ellipse using the stated information.
Vertices of the major axis at and ; length of the minor axis is 8 units.
The standard equation of the ellipse is
(Simplify your answer. Use integers or fractions for any numbers in the equation.

Question

Determine the standard equation of the ellipse using the stated information.
Vertices of the major axis at

and

; length of the minor axis is 8 units.
The standard equation of the ellipse is
(Simplify your answer. Use integers or fractions for any numbers in the equation.

UpStudy AI · Accepted Answer

)

To determine the standard equation of the ellipse, we need to use the given information about the vertices of the major axis and the length of the minor axis.

The standard equation of an ellipse with its major axis along the x-axis is given by:
$$
\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1
$$
where $$(h, k)$$ is the center of the ellipse, $$a$$ is the semi-major axis, and $$b$$ is the semi-minor axis.

Given the vertices of the major axis at $$(8,9)$$ and $$(8,-1)$$, we can determine the center $$(h, k)$$ of the ellipse. The center is the midpoint between these two points:
$$
h = \frac{8 + 8}{2} = 8
$$
$$
k = \frac{9 + (-1)}{2} = \frac{8}{2} = 4
$$
So, the center of the ellipse is $$(8, 4)$$.

Now, we need to find the lengths of the semi-major axis $$a$$ and the semi-minor axis $$b$$. The length of the major axis is the distance between the two vertices of the major axis, which is the distance between $$(8,9)$$ and $$(8,-1)$$:
$$
\text{Length of major axis} = 9 - (-1) = 10
$$
Since the semi-major axis is half the length of the major axis, we have:
$$
a = \frac{10}{2} = 5
$$

The length of the minor axis is given as 8 units. The semi-minor axis is half the length of the minor axis, so:
$$
b = \frac{8}{2} = 4
$$

Now we can write the standard equation of the ellipse using the center $$(h, k)$$ and the semi-major and semi-minor axes $$a$$ and $$b$$:
$$
\frac{(x-8)^2}{5^2} + \frac{(y-4)^2}{4^2} = 1
$$
Simplifying the denominators, we get:
$$
\frac{(x-8)^2}{25} + \frac{(y-4)^2}{16} = 1
$$

So, the standard equation of the ellipse is:
$$
\frac{(x-8)^2}{25} + \frac{(y-4)^2}{16} = 1
$$
$$ \frac{(x-8)^2}{25} + \frac{(y-4)^2}{16} = 1 $$

Determine the standard equation of the ellipse using the stated information.
Vertices of the major axis at and ; length of the minor axis is 8 units.
The standard equation of the ellipse is
(Simplify your answer. Use integers or fractions for any numbers in the equation.

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Determine the standard equation of the ellipse using the stated information. Vertices of the major axis at and ; length of the minor axis is 8 units. The standard equation of the ellipse is (Simplify your answer. Use integers or fractions for any numbers in the equation.

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Latest Geometry Questions

Determine the standard equation of the ellipse using the stated information.
Vertices of the major axis at and ; length of the minor axis is 8 units.
The standard equation of the ellipse is
(Simplify your answer. Use integers or fractions for any numbers in the equation.