Question

write the equation of the ellipse in standard form, and identify the endpoints of the major and minor axes as well as the foci.

equation in standard form:

endpoints of major axis: ( , ), ( , )

endpoints of minor axis: ( , ), ( , )

foci are at: ( , ), ( , )

question help: video ebook

Explanation:

Step1: Identify the standard form of the ellipse equation

The standard form of an ellipse equation is \(\frac{(x - h)^2}{b^2}+\frac{(y - k)^2}{a^2}=1\) (for a vertical major axis), where \((h,k)\) is the center. Given \(\frac{x^{2}}{4}+\frac{y^{2}}{25}=1\), we have \(h = 0,k = 0,a^{2}=25,a = 5,b^{2}=4,b = 2\).

Step2: Find the endpoints of the major axis

For a vertical major axis (\(a>b\)), the endpoints of the major axis are \((h,k\pm a)\). Substituting \(h = 0,k = 0,a = 5\), we get \((0,5)\) and \((0, - 5)\).

Step3: Find the endpoints of the minor axis

The endpoints of the minor axis are \((h\pm b,k)\). Substituting \(h = 0,k = 0,b = 2\), we get \((2,0)\) and \((-2,0)\).

Step4: Calculate the foci

The formula for the foci is \(c=\sqrt{a^{2}-b^{2}}\). Substitute \(a^{2}=25,b^{2}=4\), then \(c=\sqrt{25 - 4}=\sqrt{21}\). For a vertical major axis, the foci are \((h,k\pm c)\). Substituting \(h = 0,k = 0,c=\sqrt{21}\), we get \((0,\sqrt{21})\) and \((0,-\sqrt{21})\).

Answer:

Equation in standard form: \(\frac{x^{2}}{4}+\frac{y^{2}}{25}=1\)
Endpoints of major axis: \((0,5),(0, - 5)\)
Endpoints of minor axis: \((2,0),(-2,0)\)
Foci are at: \((0,\sqrt{21}),(0,-\sqrt{21})\)