Find the equation of the ellipse with the center at (2,-3), one focus at (3,-3), and one vertex at (5,-3). Also graph the equation.
the center says that the ellipse is
(x-2)^2/a^2 + (y+3)^2/b^2 = 1
Since both vertex and focus are at y=-3, the long axis is horizontal.
5-2=3, so a=3
3-2=1, so c=1
b^2+c^2 = a^2, so b^2 = 8
(x-2)^2/9 + (y+3)^2/8 = 1
graph and analysis at
http://www.wolframalpha.com/input/?i=focus%2C+vertex+for+%28x-2%29^2%2F9+%2B+%28y%2B3%29^2%2F8+%3D+1
To find the equation of an ellipse, we need to know the center, the distance from the center to a focus point, and the distance from the center to a vertex.
Given:
Center: (2, -3)
Focus: (3, -3)
Vertex: (5, -3)
First, let's find the distance from the center to the focus:
d1 = distance between (2, -3) and (3, -3) = √((3-2)² + (-3-(-3))²) = 1
Next, let's find the distance from the center to the vertex:
d2 = distance between (2, -3) and (5, -3) = √((5-2)² + (-3-(-3))²) = 3
The equation of an ellipse in standard form is:
(x-h)²/a² + (y-k)²/b² = 1
where (h, k) is the center of the ellipse, "a" is the distance from the center to a vertex, and "b" is the distance from the center to a co-vertex.
Using the given values, we can substitute them into the equation:
(x-2)²/3² + (y-(-3))²/1² = 1
Simplifying the equation further, we have:
(x-2)²/9 + (y+3)²/1 = 1
Hence, the equation of the ellipse with the given center, focus, and vertex is:
(x-2)²/9 + (y+3)²/1 = 1
To graph the ellipse, we can plot the center, focus, vertex, and any additional points to get a sense of its shape.