Question

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`?

Answer 1

Please see the explanation.

Explanation:

Standard form for the equation of an ellipse is:

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

Center: `(h,k)`

If `a > b` then:

Foci: `(h - sqrt(a^2 - b^2),k) and (h + sqrt(a^2 - b^2),k)`
Length of major axis: 2a
Length of minor axis: 2b

If `b > a` then:

Foci: `(h, k - sqrt(b^2 - a^2),k) and (h, k + sqrt(b^2 - a^2))`
Length of major axis: 2b
Length of minor axis: 2a

Put the given equation in standard form:

`(x - 1)^2/(sqrt(20))^2 + (y - -2)^2/2^2 = 1`

Center: `(1, -2)`

`sqrt(20) > 2`, therefore, first group applies:

Foci: `(1 - sqrt(20 - 4), -2) and (1 + sqrt(20 - 4), -2)`

Foci simplified: `(-3 , -2) and (5, -2)`

Length of major axis: `2sqrt(20)`
Length of minor axis: 4