Unit 12: Conic Sections
Circles, parabolas, ellipses, and hyperbolas as slices of a double cone and as loci built from a focus and directrix, in and out of the origin.
7 topics · 27 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.
Conics as Slices of a Double Cone
- How it worksHow does the angle of a flat plane slicing through a double cone determine which of the four conic sections appears?
- Common mistakeWhy isn't a parabola just a very stretched-out ellipse, and what actually makes it its own separate shape?
- Going furtherWhat happens to the cross-section when the slicing plane passes through the exact tip of the double cone?
- Going furtherWhat happens to the slice's shape as the plane's angle tilts from perfectly horizontal to parallel with the cone's side?
Your turn1 of 2
A student calls a parabola 'just an extremely elongated ellipse,' assuming it's really the same shape taken to an extreme. What's the flaw?The Parabola: Focus and Directrix
- How it worksHow does a parabola's focus-and-directrix definition keep every point equally far from both a fixed point and a fixed line?
- Worked exampleFind the focus and directrix of the parabola y2 = 12x.
- Worked exampleFind the equation of a parabola with vertex at the origin and focus at (0,3).
- Common mistakeWhy isn't the vertex of a parabola halfway between the focus and any random point on the curve?
Your turn1 of 2
Find the focus's x-coordinate of the parabola y2 = 29x.The Ellipse: Foci and Eccentricity
- How it worksHow does an ellipse's definition keep the sum of distances to its two foci constant at every point on the curve?
- Worked exampleFind the foci and eccentricity of the ellipse x2/25 + y2/9 = 1.
- Worked exampleFind the equation of an ellipse with foci at (3,0) and (-3,0) and vertices at (5,0) and (-5,0).
- Common mistakeWhy doesn't an ellipse with eccentricity close to 1 look almost like a circle, since it's actually almost flat?
Your turn1 of 2
Find the eccentricity of the ellipse x2/a2 + y2/12 = 1, as a function of a2 (where a2 > 12).The Hyperbola: Foci and Asymptotes
- How it worksHow does a hyperbola's definition keep the difference of distances to its two foci constant rather than the sum?
- Worked exampleFind the foci and asymptotes of the hyperbola x2/16 - y2/9 = 1.
- Common mistakeWhy doesn't a hyperbola's branch ever actually touch its asymptote, no matter how far out you trace it?
Your turn1 of 2
Find c (distance from center to each focus) for the hyperbola x2/24 - y2/14 = 1.Translating Conics: Center at (h, k)
- How it worksHow does replacing x with (x-h) and y with (y-k) shift a conic's graph to a new center?
- Worked exampleFind the center, vertices, and foci of the ellipse (x-2)2/16 + (y+1)2/9 = 1.
- Common mistakeWhy does completing the square on both x and y terms reveal a conic's true center hiding inside a messy equation?
- Common mistakeWhy doesn't translating a conic to a new center change its eccentricity or its overall shape?
Your turn1 of 2
A parabola with the same shape as y = 4x2 has its vertex translated to (3, 8). Write its equation.Identifying a Conic from Its Equation
- How it worksHow do the signs and coefficients on the x2 and y2 terms of a general equation identify which conic it is?
- Worked exampleFind which conic is represented by 4x2 + 9y2 - 16x + 18y - 11 = 0.
- Common mistakeWhy isn't an equation with both x2 and y2 automatically a circle?
- Ties togetherHow does identifying a conic from its x2 and y2 coefficients compare to identifying one from its eccentricity in polar form?
Your turn1 of 2
Find the ratio A/B for the equation 7x2 + 7y2 - 16x + 18y - 11 = 0 (a ratio of 1 gives a circle; otherwise it's an ellipse).Polar Equations of Conics
- How it worksHow does the eccentricity inside r = ed/(1 + e·cos(theta)) decide which conic the equation traces?
- Worked exampleFind the eccentricity and identify the conic given by r = 6/(2 + 3·cos(theta)).
- Common mistakeWhy does a polar conic's focus sit at the pole instead of at its geometric center?
- Ties togetherHow does one eccentricity value in polar form capture the same four shapes cut from slicing a cone?
Your turn1 of 2
Write r as a function of theta for the conic with eccentricity 0.618 and ed = 7 (using r = ed/(1 + e·cos(theta))).