Ellipses and Hyperbolas: Conic Sections in Precalculus

Conic Sections: Ellipses and Hyperbolas

Introduction to Conic Sections

Conic sections are curves obtained by intersecting a plane with a double-napped cone. The most common conic sections are circles, ellipses, parabolas, and hyperbolas. This section focuses on ellipses and hyperbolas, which are fundamental in Precalculus for understanding the geometry and algebra of curves.

Ellipses

Definition and Properties

• Ellipse: The set of all points in a plane such that the sum of the distances from two fixed points (called foci) is constant.

• The major axis is the longest diameter, passing through both foci.

• The minor axis is perpendicular to the major axis at the center.

• The graph of an ellipse is not a function, as it fails the vertical line test.

Standard Equations of Ellipses

• If the center is at the origin and the major axis is along the x-axis: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ where $a > b > 0$

• If the major axis is along the y-axis: $\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1$ where $a > b > 0$

• The foci are at $(\pm c, 0)$ or $(0, \pm c)$, where $c^2 = a^2 - b^2$

Derivation of the Ellipse Equation

• Let the foci be at $(\pm c, 0)$. For any point $(x, y)$ on the ellipse: $\sqrt{(x-c)^2 + y^2} + \sqrt{(x+c)^2 + y^2} = 2a$

• Through algebraic manipulation, this leads to the standard form: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$

Graphing Ellipses

• Identify the center, vertices, endpoints of the minor axis, and foci.

• Example: $4x^2 + 9y^2 = 36$

  • Divide by 36: $\frac{x^2}{9} + \frac{y^2}{4} = 1$

  • Vertices: $(\pm 3, 0)$; Minor axis endpoints: $(0, \pm 2)$

  • Foci: $c = \sqrt{9-4} = \sqrt{5}$, so $(-\sqrt{5}, 0)$ and $(\sqrt{5}, 0)$

  • Domain: $[-3, 3]$; Range: $[-2, 2]$

Ellipses Centered at (h, k)

• Horizontal major axis: $\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$

• Vertical major axis: $\frac{(x-h)^2}{b^2} + \frac{(y-k)^2}{a^2} = 1$

Applications: Reflective Property of Ellipses

• Any ray emanating from one focus reflects off the ellipse to the other focus.

• Example: Lithotripter (medical device) uses this property to focus waves.

Hyperbolas

Definition and Properties

• Hyperbola: The set of all points in a plane such that the absolute value of the difference of the distances from two fixed points (foci) is constant.

• The transverse axis passes through the vertices and foci.

• The center is the midpoint between the foci.

Standard Equations of Hyperbolas

• Transverse axis along x-axis: $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$

• Transverse axis along y-axis: $\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1$

• Foci at $(\pm c, 0)$ or $(0, \pm c)$, where $c^2 = a^2 + b^2$

• Asymptotes:

  • For $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$: $y = \pm \frac{b}{a}x$

  • For $\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1$: $y = \pm \frac{a}{b}x$

Graphing Hyperbolas

• Identify the center, vertices, foci, and asymptotes.

• Draw the fundamental rectangle using $a$ and $b$ values; asymptotes pass through its diagonals.

• Example: $\frac{x^2}{25} - \frac{y^2}{49} = 1$

  • Vertices: $(\pm 5, 0)$; Asymptotes: $y = \pm \frac{7}{5}x$

  • Foci: $c = \sqrt{25 + 49} = \sqrt{74}$

Hyperbolas Centered at (h, k)

• Horizontal transverse axis: $\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1$

• Vertical transverse axis: $\frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1$

Summary Table: Ellipses vs. Hyperbolas

Property	Ellipse	Hyperbola
Definition	Sum of distances to foci is constant	Absolute difference of distances to foci is constant
Standard Equation (centered at origin)	$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$	$\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$
Relationship of $a$, $b$, $c$	$c^2 = a^2 - b^2$	$c^2 = a^2 + b^2$
Asymptotes	None	Yes, $y = \pm \frac{b}{a}x$
Graph Shape	Oval	Two separate branches

Key Formulas

• Ellipse (center $(h, k)$): $\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$

• Hyperbola (center $(h, k)$): $\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1$

• Relationship for foci: Ellipse: $c^2 = a^2 - b^2$ Hyperbola: $c^2 = a^2 + b^2$

Example Applications

• Finding the equation of a conic given foci and vertices.

• Modeling real-world phenomena, such as the reflective property of ellipses in medical devices.