Writing Equations of Ellipses Not Centered at the Origin
Like the graphs of other equations, the graph of an ellipse can be translated. If an ellipse is translated [latex]h[/latex] units horizontally and [latex]k[/latex] units vertically, the center of the ellipse will be [latex]\left(h,k\right)[/latex]. This translation results in the standard form of the equation we saw previously, with [latex]x[/latex] replaced by [latex]\left(x-h\right)[/latex] and y replaced by [latex]\left(y-k\right)[/latex].
ellipse with a center (h,k)
The standard form of the equation of an ellipse with center [latex]\left(h,\text{ }k\right)[/latex] and major axis parallel to the x-axis is
[latex]\frac{{\left(x-h\right)}^{2}}{{a}^{2}}+\frac{{\left(y-k\right)}^{2}}{{b}^{2}}=1[/latex]
where
- [latex]a>b[/latex]
- the length of the major axis is [latex]2a[/latex]
- the coordinates of the vertices are [latex]\left(h\pm a,k\right)[/latex]
- the length of the minor axis is [latex]2b[/latex]
- the coordinates of the co-vertices are [latex]\left(h,k\pm b\right)[/latex]
- the coordinates of the foci are [latex]\left(h\pm c,k\right)[/latex], where [latex]{c}^{2}={a}^{2}-{b}^{2}[/latex].
The standard form of the equation of an ellipse with center [latex]\left(h,k\right)[/latex] and major axis parallel to the y-axis is
[latex]\frac{{\left(x-h\right)}^{2}}{{b}^{2}}+\frac{{\left(y-k\right)}^{2}}{{a}^{2}}=1[/latex]
where
- [latex]a>b[/latex]
- the length of the major axis is [latex]2a[/latex]
- the coordinates of the vertices are [latex]\left(h,k\pm a\right)[/latex]
- the length of the minor axis is [latex]2b[/latex]
- the coordinates of the co-vertices are [latex]\left(h\pm b,k\right)[/latex]
- the coordinates of the foci are [latex]\left(h,k\pm c\right)[/latex], where [latex]{c}^{2}={a}^{2}-{b}^{2}[/latex].
Just as with ellipses centered at the origin, ellipses that are centered at a point [latex]\left(h,k\right)[/latex] have vertices, co-vertices, and foci that are related by the equation [latex]{c}^{2}={a}^{2}-{b}^{2}[/latex]. We can use this relationship along with the midpoint and distance formulas to find the equation of the ellipse in standard form when the vertices and foci are given.
- Determine whether the major axis is parallel to the x– or y-axis.
- If the y-coordinates of the given vertices and foci are the same, then the major axis is parallel to the x-axis. Use the standard form [latex]\frac{{\left(x-h\right)}^{2}}{{a}^{2}}+\frac{{\left(y-k\right)}^{2}}{{b}^{2}}=1[/latex].
- If the x-coordinates of the given vertices and foci are the same, then the major axis is parallel to the y-axis. Use the standard form [latex]\frac{{\left(x-h\right)}^{2}}{{b}^{2}}+\frac{{\left(y-k\right)}^{2}}{{a}^{2}}=1[/latex].
- Identify the center of the ellipse [latex]\left(h,k\right)[/latex] using the midpoint formula and the given coordinates for the vertices.
- Find [latex]{a}^{2}[/latex] by solving for the length of the major axis, [latex]2a[/latex], which is the distance between the given vertices.
- Find [latex]{c}^{2}[/latex] using [latex]h[/latex] and [latex]k[/latex], found in Step 2, along with the given coordinates for the foci.
- Solve for [latex]{b}^{2}[/latex] using the equation [latex]{c}^{2}={a}^{2}-{b}^{2}[/latex].
- Substitute the values for [latex]h,k,{a}^{2}[/latex], and [latex]{b}^{2}[/latex] into the standard form of the equation determined in Step 1.