Question

Write an equation of an ellipse in standard form with the center at the origin and a height of 4 units and width of 5 units.

Answer 1

x^2/(25/4)+y^2/4=1
4x^2/25+y^2/4=1

Answer 2

If you don't remember the standard form, you can work it out easily enough. If the width of the ellipse is 5, and it is centered on the origin, that means (5/2,0) is a point on the ellipse, as is (-5/2,0). If the height of the ellipse is 4, and it is centered on the origin, similarly (0,2) and (0,-2) are points on the ellipse. If we remember that an ellipse is
\[\frac{x^2}{something1} + \frac{y^2}{something2} = 1\]and we have those points to work with, we can figure out the value of \(something1,2\). Let's take (0,2):
\[\frac{0^2}[something1} + \frac{2^2}{something2} = 1\]\[y^2 = something\]\[y^2 = (2)^2 = 4=something2\]Similarly for \(x\)
\[\frac{(5/2)^2}{something1} + \frac{0^2}{something2} = 1\]\[\frac{25}{4} = something1\]So our ellipse is\[\frac{x^2}{25/4} + \frac{y^2}{4} = 1\] or \[\frac{4x^2}{25} + \frac{y^2}{4} = 1\]

It's a bit more tedious to do the algebra if the origin is elsewhere, but not impossible.

Answer 3

Sorry, the line that starts with "\frac" should be
\[\frac{0^2}{something1} + \frac{2^2}{something2} = 1\]