Question

an equation of an ellipse is given.
(a) identify the center of the ellipse.
(b) identify the vertices.
(c) identify the endpoints of the minor axis.
(d) identify the foci.
(e) graph the ellipse.
express numbers in exact, simplest form.
$$ \frac { ( x - 1 ) ^ { 2 } } { 25 } + \frac { ( y + 5 ) ^ { 2 } } { 16 } = 1 $$

Explanation:

Step1: Standard form of ellipse

The standard form of an ellipse is \(\frac{(x - h)^2}{a^2}+\frac{(y - k)^2}{b^2}=1\) (\(a>b\)) or \(\frac{(x - h)^2}{b^2}+\frac{(y - k)^2}{a^2}=1\) (\(a>b\)), where \((h,k)\) is the center. For the given equation \(\frac{(x - 1)^2}{25}+\frac{(y + 5)^2}{16}=1\), we have \(h = 1,k=-5,a = 5,b = 4\)

Step2: Center of the ellipse

The center \((h,k)\) of the ellipse. Substituting \(h = 1,k=-5\), the center is \((1,-5)\)

Step3: Vertices of the ellipse

Since \(a^2=25\Rightarrow a = 5\) and the major - axis is parallel to the \(x\) - axis (because \(a^2\) is under the \(x\) - term). The vertices are \((h\pm a,k)\). Substituting \(h = 1,k=-5,a = 5\), we get \((1 + 5,-5)=(6,-5)\) and \((1-5,-5)=(-4,-5)\)

Step4: Endpoints of the minor axis

Since \(b^2=16\Rightarrow b = 4\). The endpoints of the minor axis are \((h,k\pm b)\). Substituting \(h = 1,k=-5,b = 4\), we get \((1,-5 + 4)=(1,-1)\) and \((1,-5-4)=(1,-9)\)

Step5: Foci of the ellipse

We use the formula \(c=\sqrt{a^2 - b^2}\). Substituting \(a = 5,b = 4\), we have \(c=\sqrt{25 - 16}=\sqrt{9}=3\). The foci are \((h\pm c,k)\). Substituting \(h = 1,k=-5,c = 3\), we get \((1+3,-5)=(4,-5)\) and \((1 - 3,-5)=(-2,-5)\)

Step6: Graphing the ellipse

• Plot the center \((1,-5)\)
• Plot the vertices \((6,-5)\) and \((-4,-5)\)
• Plot the endpoints of the minor axis \((1,-1)\) and \((1,-9)\)
• Sketch the ellipse passing through these four points

Answer:

(a) The center of the ellipse is \((1,-5)\)
(b) The vertices are \((6,-5)\) and \((-4,-5)\)
(c) The endpoints of the minor axis are \((1,-1)\) and \((1,-9)\)
(d) The foci are \((4,-5)\) and \((-2,-5)\)
(e) To graph: Plot center \((1,-5)\), vertices \((6,-5),(-4,-5)\), endpoints of minor axis \((1,-1),(1,-9)\) and sketch the ellipse.