# 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?

Oct 13, 2016

The center is $\left(0 , 0\right)$
The length of the major axis is $2 \sqrt{10}$
The length of the minor axis is $2 \sqrt{5}$
The foci are at $\left(0 , \sqrt{5}\right)$ and $\left(0 , - \sqrt{5}\right)$

#### 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:

${\left(y - k\right)}^{2} / {a}^{2} + {\left(x - h\right)}^{2} / {b}^{2} = 1$

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

Put the given equation in this form:

${\left(y - 0\right)}^{2} / {\left(\sqrt{10}\right)}^{2} + {\left(x - 0\right)}^{2} / {\left(\sqrt{5}\right)}^{2} = 1$

The center is $\left(0 , 0\right)$
The length of the major axis is $2 \sqrt{10}$
The length of the minor axis is $2 \sqrt{5}$
The foci are at $\left(0 , \sqrt{5}\right)$ and $\left(0 , - \sqrt{5}\right)$