10.1Exploring Ellipses
Write ellipse equations x²/a² + y²/b² = 1 in standard form. Identify center, vertices (a), co-vertices (b), and foci (c² = a² − b²). Graph and write equations from given information.
Ellipses describe planetary orbits (Kepler's first law), the shape of whispering galleries, and satellite dishes. The focus-directrix definition connects ellipses to the broader family of conic sections.
Essential Question
How does the relationship between a, b, and c define the shape and orientation of an ellipse, and how do we use the standard form equation to extract all key features?
Lesson Overview
Definition: An ellipse is the set of all points in a plane where the sum of distances to two fixed points (foci) is constant, equal to 2a.
Standard Forms:
- Horizontal major axis: x²/a² + y²/b² = 1 (a > b > 0), foci at (±c, 0), c² = a² − b²
- Vertical major axis: x²/b² + y²/a² = 1 (a > b > 0), foci at (0, ±c), c² = a² − b²
- Centered at (h, k): (x−h)²/a² + (y−k)²/b² = 1
Key Features:
- Center: (h, k)
- Vertices: endpoints of the major axis, distance a from center
- Co-vertices: endpoints of the minor axis, distance b from center
- Foci: two fixed interior points, distance c from center
Eccentricity: e = c/a (0 < e < 1 for ellipses; closer to 0 = more circular, closer to 1 = more elongated)
Key relationship: c² = a² − b² (always: a > b, a > c)
Worked Examples
Identify all features of x²/25 + y²/9 = 1.
a² = 25 → a = 5; b² = 9 → b = 3
c² = 25 − 9 = 16 → c = 4
Horizontal major axis (larger denominator under x²)
Center: (0,0); Vertices: (±5, 0); Co-vertices: (0, ±3); Foci: (±4, 0)
Write the equation of an ellipse with vertices at (±6, 0) and co-vertices at (0, ±4).
a = 6 → a² = 36; b = 4 → b² = 16
Horizontal major axis
Equation: x²/36 + y²/16 = 1
Find all features of (x−2)²/16 + (y+3)²/4 = 1.
Center: (2, −3); a² = 16 → a = 4; b² = 4 → b = 2
c² = 16 − 4 = 12 → c = 2√3
Horizontal major axis
Vertices: (2±4, −3) = (6,−3) and (−2,−3)
Co-vertices: (2, −3±2) = (2,−1) and (2,−5)
Foci: (2±2√3, −3)
Write the equation of an ellipse with foci at (0, ±3) and vertices at (0, ±5).
Vertical major axis (foci on y-axis)
a = 5, c = 3 → b² = a² − c² = 25 − 9 = 16
Equation: x²/16 + y²/25 = 1
Convert 4x² + 9y² − 16x + 18y − 11 = 0 to standard form.
Group: (4x² − 16x) + (9y² + 18y) = 11
Factor: 4(x² − 4x) + 9(y² + 2y) = 11
Complete the square: 4(x−2)² − 16 + 9(y+1)² − 9 = 11
4(x−2)² + 9(y+1)² = 36
Divide by 36: (x−2)²/9 + (y+1)²/4 = 1
Guided Practice
Identify all features of x²/49 + y²/36 = 1.
Hint: Which denominator is larger? That determines the major axis direction. Then find c² = a² − b².
Write the equation of an ellipse with vertices at (0, ±7) and foci at (0, ±√33).
Hint: Vertical major axis. Use b² = a² − c² to find b².
Find the center, vertices, and foci of (x+1)²/25 + (y−4)²/9 = 1.
Hint: Center is (h, k) = (−1, 4). Then apply the standard formulas with a² = 25, b² = 9.
An ellipse has center (0,0), a vertex at (4, 0), and passes through (2, √3·b/2). Find the equation.
Hint: Use a = 4. Substitute the point to find b².
Convert 9x² + 4y² − 54x + 8y + 49 = 0 to standard form.
Hint: Group x and y terms, factor out coefficients, complete the square for each variable.
Key Vocabulary
Ellipse
Set of all points where the sum of distances to two fixed points (foci) equals 2a.
Major axis
The longer axis of the ellipse; length 2a.
Minor axis
The shorter axis of the ellipse; length 2b.
Vertices
Endpoints of the major axis, distance a from center.
Co-vertices
Endpoints of the minor axis, distance b from center.
Foci (singular: focus)
Two fixed points inside the ellipse; c² = a² − b².
Eccentricity (e)
e = c/a; measures how 'stretched' the ellipse is (0 < e < 1).
Standard form
(x−h)²/a² + (y−k)²/b² = 1 (horizontal) or (x−h)²/b² + (y−k)²/a² = 1 (vertical).
Practice Quiz
Interactive Practice — 5 Questions
For x²/16 + y²/9 = 1, the foci are at:
An ellipse has a = 5 and b = 3. Its eccentricity is:
Which equation has a vertical major axis?
For (x−3)²/25 + (y+2)²/16 = 1, the center is:
The relationship between a, b, and c for an ellipse is:
Independent Practice
Independent Practice
Find all features of x²/100 + y²/64 = 1.
Write the equation of an ellipse with vertices (±8, 0) and foci (±5, 0).
Find all features of (x+3)²/36 + (y−1)²/11 = 1.
Write the equation of an ellipse with center (2, −1), vertical major axis, a = 6, b = 4.
Convert 25x² + 4y² + 100x − 40y + 100 = 0 to standard form and identify all features.
Common Mistakes
Assuming the larger denominator always goes under x²
The larger denominator determines the major axis direction. Under x² → horizontal; under y² → vertical.
Using c² = a² + b² (like the Pythagorean theorem)
For ellipses: c² = a² − b². The foci are INSIDE the ellipse, so c < a.
Confusing vertices with foci
Vertices are the endpoints of the major axis (distance a). Foci are interior points (distance c). Always c < a.
Forgetting to divide by the constant when completing the square
After completing the square, divide EVERY term by the constant on the right to get the standard form = 1.
Math Tips
Quick check: in standard form, the larger denominator tells you the major axis direction. Larger under x² → horizontal; larger under y² → vertical.
Memory trick for c² = a² − b²: think of a right triangle inside the ellipse with hypotenuse a, leg b, and leg c.
Eccentricity e = c/a is always between 0 and 1. Near 0 = nearly circular; near 1 = very elongated (like a comet orbit).
When completing the square, factor out the leading coefficient BEFORE completing the square: 4(x² − 4x + 4) = 4(x−2)².
Always verify: a > b > 0 and a > c > 0. If these don't hold, recheck which value is a and which is b.