(x-2)^2 / 49 + (y+1)^2 / 25 = 1
a.) what are the vertices?
b.) What are the major and minor axises?
I always have troubling understanding how to find these. :(
Any help is greatly appreciated!
for x^2/a^2 + y^2/b^2 = 1,
semi-major axes are a and b. Select the larger for major, smaller for minor.
so, here major=14, minor=10
foci are on the major axis, at distance c from center, where a^2 = b^2 + c^2
so, c = √(49-25) = √24 = 2√6
Knowing the distances, and the center, and the direction of the major axis, you can easily write the coordinates of the foci and vertices.
Thank you, @Steve !
Would you happen to know how to find the Foci for this equation?!?!
:)
To find the vertices and major/minor axes of an ellipse, you can follow these steps:
Step 1: Rewrite the given equation in the standard form of an ellipse equation:
(x - h)^2 / a^2 + (y - k)^2 / b^2 = 1
where (h, k) is the center of the ellipse, a is the length of the semi-major axis, and b is the length of the semi-minor axis.
Comparing the given equation, (x-2)^2 / 49 + (y+1)^2 / 25 = 1, with the standard form, we can determine the values of h, k, a, and b.
Step 2: Identify the values of h, k, a, and b.
Comparing the given equation with the standard form, we can see that:
h = 2 (the x-coordinate of the center)
k = -1 (the y-coordinate of the center)
a^2 = 49 (the square of the semi-major axis)
b^2 = 25 (the square of the semi-minor axis)
Step 3: Find the coordinates of the vertices.
The vertices of the ellipse can be found by adding or subtracting the length of the semi-major axis (a) from the center (h, k). In this case, since a^2 = 49, a = 7.
Therefore, the coordinates of the vertices are:
Vertex 1: (h + a, k) = (2 + 7, -1) = (9, -1)
Vertex 2: (h - a, k) = (2 - 7, -1) = (-5, -1)
So the vertices are (9, -1) and (-5, -1).
Step 4: Find the lengths of the major and minor axes.
The major axis is the line segment passing through the center and the vertices. So, the length of the major axis is the distance between the two vertices:
Length of the major axis = |9 - (-5)| = 14
The minor axis is the line segment perpendicular to the major axis and passing through the center. So, the length of the minor axis is twice the length of the semi-minor axis:
Length of the minor axis = 2 * √b^2 = 2 * √25 = 10
Therefore,
a.) The vertices of the ellipse are (9, -1) and (-5, -1).
b.) The length of the major axis is 14 and the length of the minor axis is 10.