Equation of Each Ellipse and Finding the Foci, Vertices, and Co– Vertices of Ellipses

• If major axis of an ellipse is parallel to \(x\), its called horizontal ellipse.
• If major axis of an ellipse is parallel to \(y\), its called vertical ellipse.

Step by Step Guide to Find Equation of Ellipses

• Horizontal: \(\color{blue}{\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1}\)
• Vertical: \(\color{blue}{\frac{(x-h)^2}{b^2}+\frac{(y-k)^2}{a^2}=1}\)

Equation of Each Ellipse and Finding the Foci, Vertices, and Co– Vertices of Ellipses – Example 1:

Solution:

The standard form of the equation of an Ellipse is: \(\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1\)

Then, \((h=2, k=-4, a=6, b=4)\).

So, the center is \((2, -4)\).

The vertices are \((h+a, k), (h-a, k) →(8, 4), (-4, 4)\)

Evaluate \(c\): \(c=\sqrt{a^2-b^2}\) \(=\sqrt{36-16}=2\sqrt{5}\)

Then the foci are \((2+2\sqrt{5}, -4)\) and \((2-2\sqrt{5}, -4)\).

Exercises for Equation of Finding the Foci, Vertices, and Co– Vertices of Ellipses

Find the center, vertices, and foci of each ellipse.

• \(\color{blue}{9x^2+4y^2=1}\)
• \(\color{blue}{16x^2+25y^2=100}\)
• \(\color{blue}{25x^2+4y^2+100x-40y=400}\)
• \(\color{blue}{\frac{(x-1)^2}{9}+\frac{y^2}{5}=100}\)

• \(\color{blue}{Center: (0, 0), Vertices: (0,\frac{1}{2}), (0, -\frac{1}{2}), foci: (0, \frac{\sqrt{5}}{6}), (0, -\frac{\sqrt{5}}{6})}\)
• \(\color{blue}{Center: (0, 0), Vertices: (\frac{5}{2}, 0), (-\frac{5}{2}, 0), foci: (\frac{3}{2}, 0), (-\frac{3}{2}, 0)}\)
• \(\color{blue}{Center: (-2, 5), Vertices: (-2,5+5\sqrt{6}), (-2, 5-5\sqrt{6}), foci: (-2, 5+3\sqrt{14}), (-2, 5-3\sqrt{14})}\)
• \(\color{blue}{Center: (1, 0), Vertices: (31, 0), (-29, 0), foci: (21, 0), (-19, 0)}\)

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