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Equations of Ellipses
& Hyperbolas

Derive equations from the distance definition: the sum or difference of distances from two foci is constant.

Geometry

What Are Conic Sections?

Conic sections are curves formed by slicing a double cone with a plane. Depending on the angle of the cut, you get a circle, ellipse, parabola, or hyperbola. Today we focus on ellipses and hyperbolas: both are defined by two special points called foci.

Ellipse
Sum of distances from foci is constant
Hyperbola
Difference of distances from foci is constant

Key Vocabulary

Foci \(F_1, F_2\)
Two fixed points that define the curve
Center
Midpoint between the two foci
Vertices
Points where the curve crosses its axes
\(a\) (semi-major)
Half the longest axis of an ellipse, or half the transverse axis of a hyperbola
\(b\) (semi-minor)
Half the shortest axis of an ellipse, or half the conjugate axis
\(c\) (focal distance)
Distance from center to each focus

The Distance Definition

\(F_1\) \(F_2\) \(P\) \(d_1\) \(d_2\) \(C\) \(a\) \(c\)

Deriving the Ellipse Equation

1
Start with the definition: \(\sqrt{(x+c)^2 + y^2} + \sqrt{(x-c)^2 + y^2} = 2a\)
2
Isolate one radical and square both sides. Expand and simplify.
3
Square again to eliminate the remaining radical. Collect terms.
4
Define \(b^2 = a^2 - c^2\). Divide both sides by \(a^2 b^2\).

Anatomy of \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\)

\((a, 0)\) \((-a, 0)\) \((0, b)\) \((0, -b)\) \((-c, 0)\) \((c, 0)\) \(a\) \(b\)

The Hyperbola Definition

\(F_1\) \(F_2\) \(P\) \(d_1\) \(d_2\)

Deriving the Hyperbola Equation

1
Start with: \(|\sqrt{(x+c)^2 + y^2} - \sqrt{(x-c)^2 + y^2}| = 2a\)
2
Same algebraic technique: isolate one radical, square, simplify, square again.
3
Define \(b^2 = c^2 - a^2\) (note the reversed subtraction from the ellipse).

Ellipse vs Hyperbola

Ellipse Hyperbola
Definition \(d_1 + d_2 = 2a\) \(|d_1 - d_2| = 2a\)
Standard Form \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\)
Key Sign + (plus) - (minus)
Relationship \(c^2 = a^2 - b^2\) \(c^2 = a^2 + b^2\)
Shape Closed curve Two open branches
Foci location Inside the curve Inside each branch

How to Derive the Equation

1
Identify the foci \(F_1\) and \(F_2\). Find \(c\) (distance from center to focus).
2
Determine the constant: is it a sum (\(2a\) for ellipse) or difference (\(2a\) for hyperbola)?
3
Find \(a\) from the constant: \(a = \frac{\text{constant}}{2}\).
4
Find \(b\) using the relationship: \(b^2 = a^2 - c^2\) (ellipse) or \(b^2 = c^2 - a^2\) (hyperbola).
5
Write the equation. For an ellipse, \(a^2\) is the larger denominator. For a hyperbola, \(a^2\) is under the positive term.

Example 1: Ellipse from Foci

Problem: An ellipse has foci at \((\pm 3, 0)\) and the sum of distances from any point on the ellipse to the foci is 10. Find the equation.

1
Foci: \(c = 3\) (center at origin, horizontal major axis).
2
Constant sum: \(2a = 10\), so \(a = 5\).
3
Find \(b\): \(b^2 = a^2 - c^2 = 25 - 9 = 16\), so \(b = 4\).

Example 2: Hyperbola from Foci

Problem: A hyperbola has foci at \((\pm 5, 0)\) and the absolute difference of distances is 6. Find the equation.

1
Foci: \(c = 5\) (center at origin, opens left-right).
2
Constant difference: \(2a = 6\), so \(a = 3\).
3
Find \(b\): \(b^2 = c^2 - a^2 = 25 - 9 = 16\), so \(b = 4\).

Example 3: Vertical Ellipse

Problem: An ellipse has foci at \((0, \pm 4)\) and the sum of distances is 12. Derive the equation.

\(F_1(0, 4)\) \(F_2(0, -4)\) \((0, 6)\) \((0, -6)\)

Vertical Hyperbola Orientation

When the foci are on the y-axis at \((0, \pm c)\), the transverse axis is vertical. The standard form changes:

Horizontal (foci on x-axis)
$$\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$$
Vertical (foci on y-axis)
$$\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1$$

Common Mistakes

✕ Mixing up the \(b^2\) formula
For ellipses: \(b^2 = a^2 - c^2\). For hyperbolas: \(b^2 = c^2 - a^2\). Using the wrong one gives negative values.
✕ Confusing + and -
An ellipse has \(+\) between the fractions. A hyperbola has \(-\). This is the single most important difference.
✕ Wrong axis for \(a^2\)
For an ellipse, \(a^2\) is the larger denominator. For a hyperbola, \(a^2\) is under the positive term, regardless of size.
✕ Forgetting \(2a\), not \(a\)
The constant sum or difference is \(2a\), not \(a\). Always divide the given constant by 2.

Question 1 of 5

An ellipse has foci at \((\pm 4, 0)\) and the sum of distances from any point on the ellipse to the foci is 10. What is the equation?

Question 2 of 5

A hyperbola has foci at \((\pm 13, 0)\) and vertices at \((\pm 5, 0)\). What is \(b^2\)?

Question 3 of 5

Which equation represents a hyperbola?

Question 4 of 5

An ellipse has foci at \((0, \pm 3)\) and the sum of distances is 10. Which equation is correct?

Question 5 of 5

For the ellipse \(\frac{x^2}{49} + \frac{y^2}{24} = 1\), what are the foci?