PRE-CAL | Intro to Conic Sections, Circles, & Ellipses

Conic Section

• curves obtained by the intersection of a plan and a double cone with two identical nappes/cone (one part of a double cone)

Circle

• occurs when the plane is horizontal

Ellipses

• occurs when the (tilted) plane intersects only one cone to form a bounded curve.

• looks like an oval/oblong

Parabola

• occurs when the (tilted) plane intersects only one cone to form an unbounded curve.

Hyperbola

• when the plane intersects both cones to form two unbounded curves, also known as branches

Circles

• A graph where all the points are equidistant from a fixed point called the center.

Equation

• (x-h)² + (y-k)² = r²

  • (x, y) are the coordinates of the variable points on the actual graph.

  • (h,k) are the coordinates of the fixed point of the circle.

  • r is the radius, which is the distance of a variable point to the fixed point.

Graphing (using the standard equation)

1. Identify the center of the circle and graph it based on the constants in the equation.

2. Identify the radius, or r, by getting the square root of the value on the right side of the equation.

3. Identify and graph the 4 points by moving r units up, down, left, and right from the center.

4. Connect all the points properly to form a circular shape.

Semicircle Equations

Derivation

• To derive the equations, you must solve for a certain variable depending on how the semicircle opens.

  • If it is the circle’s the upper half or lower half, solve for y.

  • If it is the circle’s left half or right half, solve for x.

Equations & Graphs

Note: Graphs have the center at the origin and a radius at 5.

Upper Half

Lower Half

Right Half

Left Half

Graphing (using the standard equation)

• Follow the same rules for graphing circles but only graph the half identified based on the equation.

Ellipse

• a set of points wherein their respective sums of the distances from two fixed points are equal or constant.

Properties

• Focus: the fixed points described in the definition above.

• Center: the fixed point that is the point of symmetry of the ellipse.

• Minor axis: the shortest diameter and axis of symmetry of the ellipse.

• Major axis: the longest diameter and axis of symmetry of the ellipse.

• Vertices: the points on the ellipse that are the endpoints of the major axis.

• Co-vertices: the points on the ellipse that are the endpoints of the minor axis.

Formula

Variables

• a is the distance from the center to a vertex

• 2a is the length of the major axis.

• b is the distance form the center to a co-vertex

• 2b is the length of the minor axis

• c is the distance from the center to the focus.

Pythagorean Relationship

• a is equal to the line segment connecting a focus and a co-vertex, forming the hypotenuse of a right triangle.

• a² = b² + c²

• c² = a² - b²

General Equation

• |PF₁| + |PF₂| = 2a

  • PF is the distance between a point on the ellipse and the focus.

  • a is the distance from the center to one end point of the major axis.

Standard Equation for HORIZONTAL ELLIPSES

• (x-h)²/a² + (y-k)²/b² = 1; wherein a > b

Standard Equation for VERTICAL ELLIPSES

• (x-h)²/b² + (y-k)²/a² = 1; wherein a > b

Graphing an Ellipse

Latus Rectum

• a chord passing through a focus, is perpendicular to the major axis, and has its endpoints located on the ellipse

• Length is equal to 2b²/a units.

How to Graph an Ellipse Using the Standard Equation

1. Identify the center of the ellipse and graph it based on the constants in the equation.

2. Identify and graph the first two points by getting the square root of the denominator of the first term.

   • Identify the coordinates of the endpoints by adding and subtracting this value from the x-coordinate of the center.

   • The y-coodinate stays the same.

   • Graph the points on the Cartesian plane.

     • In a horizontal ellipse, it should form the major axis. The graphed points would have been the vertices.

     • In a vertical ellipse, it should form the minor axis. The graphed points would have been the co-vertices.

3. Identify and graph the next two points by getting the square root of the denominator of the second term.

   • Identify the coordinates of the endpoints by adding and subtracting this value from the y-coordinate of the center.

   • The x-coodinate stays the same.

   • Graph the points on the Cartesian plane.

     • In a horizontal ellipse, it should form the minor axis.

     • In a vertical ellipse, it should form the major axis.

4. Identify the value of c by getting the square root of a² - b².

   • Identify the foci.

     • In a horizontal ellipse, add and subtract the solved value from the x-coordinate of the center.

     • In a vertical ellipse, add and subtract the solved value from the y-coordinate of the center.

5. Identify the end points of latera recta.

   • Get the distance of the foci to the endpoint of a latus rectum by solving for b²/a.

   • Graph the points on the Cartesian plane.

     • In a horizontal ellipse, add and subtract the solved value from the y-coordinate of the foci.

     • In a vertical ellipse, add and subtract the solved value from the x-coordinate of the foci.

6. Connect all the points properly to form an oblong shape.