Question

How do you find the coordinates of the center, foci, the length of the major and minor axis given `y^2/10+x^2/5=1`?

Answer 1

The center is `(0,0)`
The length of the major axis is `2sqrt(10)`
The length of the minor axis is `2sqrt(5)`
The foci are at `(0, sqrt(5))` and `(0, -sqrt(5))`

Explanation:

Please observe the denominator of the y term (10) is greater than the denominator of the x term (5). The standard form for an ellipse of this type is:

`(y - k)^2/a^2 + (x - h)^2/b^2 = 1`

The center of the ellipse is the point `(h, k)`.
The value of `a` is the length of the semi-major axis and `2a` is the length of the major axis
The value of `b` is the length of the semi-minor axis and `2a` is the length of the minor axis.
The focal distance is, `c = sqrt(a^2 - b^2)`
The foci are located at `(h, k + c)` and `(h, k - c)`

Put the given equation in this form:

`(y - 0)^2/(sqrt(10))^2 + (x - 0)^2/(sqrt(5))^2 = 1`

The center is `(0,0)`
The length of the major axis is `2sqrt(10)`
The length of the minor axis is `2sqrt(5)`
The foci are at `(0, sqrt(5))` and `(0, -sqrt(5))`