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Unit 12: Conic Sections

Circles, parabolas, ellipses, and hyperbolas as literal slices through a cone, each with its own defining equation.

6 topics · 21 questions

Each topic is a path, not a menu: work down it in order and every question builds on the one above. Click any of them to watch the answer draw itself.

Conic Sections as Slices of a Cone

  1. How it worksHow does the angle of the slicing plane through a cone decide whether you get a circle, ellipse, parabola, or hyperbola?
  2. Going furtherWhat happens to the resulting conic when the slicing plane passes right through the cone's very tip?
  3. Common mistakeWhy isn't a conic section defined by its shape on paper rather than by how the plane cuts the cone?
Your turn1 of 2
A plane slices through a double cone at an angle STEEP enough to cut through BOTH nappes (both halves) of the cone. What conic section results?

Circles in Center-Radius Form

  1. How it worksHow does the distance formula turn into the standard equation of a circle?
  2. Worked exampleFind the equation of the circle with center (3, -2) and radius 5, then find the center and radius from (x-3)2 + (y+2)2 = 25.
  3. Common mistakeWhy doesn't the radius appear as r in the circle's equation instead of as r squared?
Your turn1 of 2
For the circle (x - 7)2 + (y - 6)2 = 93, solve for (y - 6)2 as a function of x.

Completing the Square to Find a Circle's Center and Radius

  1. How it worksHow does completing the square on both x and y terms reveal a circle's center-radius form?
  2. Worked exampleFind the center and radius of the circle x2 + y2 - 6x + 4y - 3 = 0 by completing the square.
  3. Worked exampleFind the center and radius of the circle x2 + y2 + 8x - 2y + 8 = 0 by completing the square.
Your turn1 of 2
Find the radius of the circle x2 + y2 + 7x - 7y + -2 = 0.

Parabolas Defined by a Focus and Directrix

  1. How it worksHow does the distance from a point to the focus matching the distance to the directrix generate a parabola's curve?
  2. Worked exampleFind the focus and directrix of the parabola y = (1 over 8)x2.
  3. Common mistakeWhy isn't the vertex of a parabola the same point as its focus?
  4. Ties togetherHow does a parabola's focus-directrix definition connect to why satellite dishes are shaped like parabolas?
Your turn1 of 2
A parabola has focus at (0, 7) and vertex at the origin. Write its equation y(x).

Ellipses and Their Two Foci

  1. How it worksHow does the constant sum of distances to two foci define every point on an ellipse?
  2. Worked exampleFind the foci and vertices of the ellipse x2 over 25 plus y2 over 9 equals 1.
  3. Common mistakeWhy doesn't an ellipse have just one center-like point the way a circle does?
  4. Going furtherWhat happens to an ellipse's shape as its two foci move closer and closer together toward the center?
Your turn1 of 2
For the ellipse x2/38 + y2/7 = 1, solve for y2 as a function of x.

Hyperbolas and Their Two Branches

  1. How it worksHow does the constant difference of distances to two foci, instead of the sum, generate a hyperbola's two branches?
  2. Worked exampleFind the foci and asymptotes of the hyperbola x2 over 16 minus y2 over 9 equals 1.
  3. Common mistakeWhy doesn't a hyperbola's graph ever actually touch its own asymptotes?
  4. Ties togetherHow does swapping a sum for a difference turn an ellipse's defining equation into a hyperbola's?
Your turn1 of 2
For the hyperbola x2/15 - y2/14 = 1, solve for y2 as a function of x.