Finding the Equation of an Ellipse Find an equation of the ellipse that has foci (-3, 2) and (5, 2) and that has a major axis of length 10.

Question

Finding the Equation of an Ellipse

Find an equation of the ellipse that has foci (-3, 2) and (5, 2) and that has a major axis of length 10.

RECALL
The midpoint of the segment joining [latex](x_1,y_1)[/latex] and [latex](x_2,y_2)[/latex] is

[latex]\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2} \right) [/latex]

Accepted Answer

Because the foci (-3, 2) and (5, 2) lie on the horizontal line y = 2 the ellipse is a horizontal ellipse. The center of the ellipse is the midpoint of the line segment joining the foci. Using the midpoint formula yields

[latex]h=\frac{-3+5}{2}=1 \quad ext { and } \quad k=\frac{2+2}{2}=2.[/latex]

The center of the ellipse is (1, 2) See Figure 17. Because the length of the major axis is 10, the vertices must be at a distance a = 5 units from the center.
In Figure 17, the foci are four units from the center. Thus, [latex] c=4 . ext { Now find } b^{2} :[/latex]

[latex] b^{2}=a^{2}-c^{2}=(5)^{2}-(4)^{2}=9[/latex]

Because the major axis is horizontal, the standard form of the equation of the ellipse is

[latex]\begin{array}{ll}\frac{(x-h)^{2}}{a^{2}}+\frac{(y-k)^{2}}{b^{2}}=1 & a>b>0 \\frac{(x-1)^{2}}{25}+\frac{(y-2)^{2}}{9}=1 & ext { Replace } h ext { with } 1, k ext { with } 2, \ &a^{2} ext { with } 25, ext { and } b^{2} ext { with } 9 .\end{array}[/latex]

For this horizontal ellipse with center [latex] (1,2) ext {, we have } a=5, b=3 ext {, and } c=4 ext {. }[/latex]

Vertices:

[latex](h \pm a, k)=(1 \pm 5,2)=(-4,2) ext { and }(6,2)[/latex]

Endpoints of Minor Axis:

[latex](h, k \pm b)=(1,2 \pm 3)=(1,-1) ext { and }(1,5)[/latex]

Using the center and these four points we sketch the graph shown in Figure 18.

Question 11.3.3: Finding the Equation of an Ellipse Find an equation of the e...

Related Answered Questions

Verified Answer:

Verified Answer:

Finding the Equation of a Hyperbola Find the standard form of the equation of a hyperbola with vertices (±4, 0) and foci (±5, 0). ...

Verified Answer:

Graphing a Parabola Find the vertex, focus, and directrix of the parabola 2y² – 8y – x + 7 = 0. Sketch the graph of the parabola. ...

Verified Answer:

Finding the Equation of a Parabola Find the standard equation of a parabola with vertex (0, 0) satisfying the given description. a. The focus is(-3, 0). b. The axis of the parabola is the y-axis, and the graph passes through the point (-4, 2). ...

Verified Answer:

Calculating Some Properties of the Hubble Space Telescope The parabolic mirror used in the Hubble Space Telescope has a diameter of 94.5 inches. See Figure 12. Find the equation of the parabola if its focus is 2304 inches from the vertex.What is the thickness of the mirror at the edges? ...

Verified Answer:

Finding an Equation of an Ellipse Find the standard form of the equation of the ellipse that has vertex (5, 0) and foci (±4, 0). ...

Verified Answer:

Graphing an Ellipse Sketch a graph of the ellipse whose equation is 9x² + 4y² = 36. Find the foci of the ellipse. ...

Verified Answer:

Converting to Standard Form Find the center, vertices, and foci of the ellipse with equation 3x² + 4y² + 12x – 8y – 32 = 0. ...

Verified Answer:

Lithotripsy An elliptical water tank has a major axis of length 6 feet and a minor axis of length 4 feet. The source of high-energy shock waves from a lithotripter is placed at one focus of the tank. To smash the kidney stone of a patient, how far should the stone be positioned from the source? ...

Verified Answer: