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:
Tutor-style math help
Equation of Each Ellipse and Finding the Foci, Vertices, and Co– Vertices of Ellipses: what to notice and how to work it
Conics skill
Conic sections are graph shapes with standard forms. The equation tells you the center or vertex, then the key distances shape the graph.
What to notice first
For an ellipse, the larger denominator marks the major axis direction.
Common student mistake
Do not read signs backward in \((x-h)\) and \((y-k)\). The center or vertex uses \(h\) and \(k\), not the visible sign alone.
Key formulas and cues
\((x-h)^2+(y-k)^2=r^2\)
\(\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1\)
\((y-k)=a(x-h)^2\)
\(\frac{(x-h)^2}{a^2}-\frac{(y-k)^2}{b^2}=1\)
A reliable path
- Match the formIdentify the conic by its equation pattern.
- Read featuresFind the center, vertex, radius, axes, foci, or asymptotes.
- Sketch from anchorsPlot key points first, then draw the curve.
Worked examples
Circle center and radius
Example: \((x-4)^2+(y+1)^2=25\)
- Compare to circle standard form.
- The center is (4, -1).
- The radius is the square root of 25.
Answer: Center (4, -1), radius 5
Parabola direction
Example: \((x-2)^2=8(y+3)\)
- The x part is squared.
- The parabola opens up or down.
- The positive coefficient means it opens up.
Answer: Opens up
Try one before moving on
Try: Find the center of \((x+3)^2+(y-2)^2=16\).
Answer: (-3, 2).
Next step: do the matching worksheet or quiz while the method is still fresh, then come back and explain the first step in your own words.
x
Equation of Each Ellipse and Finding the Foci, Vertices, and Co– Vertices of Ellipses: pop-up practice
Answer these quick questions, then use the feedback to decide which part of the lesson to review.
Choose an answer to begin.
1. \((x-1)^2+(y+2)^2=9\) has center:
2. A hyperbola usually has:
3. An ellipse has:
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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