for my homework problem i have the equation:
(x^2/16) + (y^2/1) = 1
I need to find out the center, the vertices, co- vertices and the Foci.
I have trouble knowing how to find them, so if you can please explain them.
Thank you.
To find the center, vertices, co-vertices, and foci of the ellipse given by the equation:
(x^2/16) + (y^2/1) = 1
Let's start by re-writing the equation in the standard form of an ellipse:
(x^2/a^2) + (y^2/b^2) = 1
Where "a" is the distance from the center to the vertices along the major axis, and "b" is the distance from the center to the co-vertices along the minor axis.
Comparing this with the given equation, we can see that a^2 = 16 and b^2 = 1.
Center:
The center of the ellipse is given by the coordinates (h, k), where h is the x-coordinate and k is the y-coordinate. In this case, we can see that the x^2 term is divided by 16 and the y^2 term is divided by 1, which means that the center is at the origin (0,0).
Vertices:
The vertices are the points on the ellipse where the major axis intersects the ellipse. Since a^2 = 16, we can take the square root of 16 to find "a" which is 4. The vertices along the x-axis are (±a, 0), which in this case becomes (±4, 0). So the vertices of the ellipse are (4, 0) and (-4, 0).
Co-vertices:
The co-vertices are the points on the ellipse where the minor axis intersects the ellipse. Since b^2 = 1, we can take the square root of 1 to find "b" which is 1. The co-vertices along the y-axis are (0, ±b), which in this case becomes (0, ±1). So the co-vertices of the ellipse are (0, 1) and (0, -1).
Foci:
To find the foci of the ellipse, we can use the equation c^2 = a^2 - b^2, where c is the distance from the center to the foci along the major axis. From our previous calculations, we know that a = 4 and b = 1. Plugging these values into the equation, we get c^2 = 16 - 1, which simplifies to c^2 = 15. Taking the square root of 15, we find c ≈ 3.87.
The foci along the x-axis are given by (±c, 0), which in this case becomes (±3.87, 0). So the foci of the ellipse are approximately (3.87, 0) and (-3.87, 0).
To summarize:
- The center of the ellipse is (0, 0).
- The vertices of the ellipse are (4, 0) and (-4, 0).
- The co-vertices of the ellipse are (0, 1) and (0, -1).
- The foci of the ellipse are approximately (3.87, 0) and (-3.87, 0).