# How do you find the coordinates of the center, foci, lengths of the major and minor axes given (x-1)^2/20+(y+2)^2/4=1?

Nov 19, 2016

#### Explanation:

Standard form for the equation of an ellipse is:

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

Center: $\left(h , k\right)$

If $a > b$ then:

Foci: $\left(h - \sqrt{{a}^{2} - {b}^{2}} , k\right) \mathmr{and} \left(h + \sqrt{{a}^{2} - {b}^{2}} , k\right)$
Length of major axis: 2a
Length of minor axis: 2b

If $b > a$ then:

Foci: $\left(h , k - \sqrt{{b}^{2} - {a}^{2}} , k\right) \mathmr{and} \left(h , k + \sqrt{{b}^{2} - {a}^{2}}\right)$
Length of major axis: 2b
Length of minor axis: 2a

Put the given equation in standard form:

${\left(x - 1\right)}^{2} / {\left(\sqrt{20}\right)}^{2} + {\left(y - - 2\right)}^{2} / {2}^{2} = 1$

Center: $\left(1 , - 2\right)$

$\sqrt{20} > 2$, therefore, first group applies:

Foci: $\left(1 - \sqrt{20 - 4} , - 2\right) \mathmr{and} \left(1 + \sqrt{20 - 4} , - 2\right)$

Foci simplified: $\left(- 3 , - 2\right) \mathmr{and} \left(5 , - 2\right)$

Length of major axis: $2 \sqrt{20}$
Length of minor axis: 4