Conic Sections
Learn how distances define conic sections.
Use equations to find key points and sketch each curve.
Parabolas · Ellipses · Hyperbolas
Definitions of the three conics
Intersecting a cone with a plane produces a conic. We can also define each conic using distances.
Parabola: equal distances
Every point on a parabola is equally far from a fixed point (the focus) and a fixed line (the directrix).
Ellipse: fixed sum
Add the distances from a point on the ellipse to the two foci. The total is the same everywhere on the ellipse.
Hyperbola: fixed difference
Subtract the shorter distance to a focus from the longer one. The difference is the same everywhere on the hyperbola.
Parabola: equal distances to a focus and directrix
Press Play to move P along the parabola. Its distance to the focus equals its perpendicular distance to the directrix.
Vertex at the origin
Focus $(0,p)$ · directrix $y=-p$
Vertex $(0,0)$ · axis $x=0$
The vertex is halfway between the focus and directrix. $p>0$ opens upward; $p<0$ opens downward.
Derive the equation from equal distances
The control excludes p = 0, which gives the line x = 0 instead of a parabola.
Ellipse: the distances to the foci add to 2a
Press Play to move P along the ellipse. The two focal distances change, but their sum always equals 2a.
Center at the origin
Foci $(\pm c,0)$ · vertices $(\pm a,0)$
Center $(0,0)$ · major axis $y=0$
The semi-major axis is $a$, semi-minor axis is $b$, and $c^2=a^2-b^2$ ($c<a$). Distance sum: $PF_1+PF_2=2a$.
Derive the equation from distance sum
Use a right triangle to find c
The top point is equally far from both foci. Since the two distances add to 2a, each distance is a.
$c$ measures the distance from the center to either focus.
For $a=5$ and $b=3$, $c=\sqrt{25-9}=4$.
Find the ellipse’s major axis
First divide so the right-hand side equals 1. The larger denominator identifies the major axis.
The larger denominator is under $y^2$, so the major axis is vertical.
$a=5$, $b=3$, $c=4$.
- Vertices: $(0,\pm5)$
- Foci: $(0,\pm4)$
- Minor-axis endpoints: $(\pm3,0)$
Blue: vertices · orange: foci · teal: minor-axis endpoints.
Hyperbola: the distance difference is 2a
Press Play to move P along either branch. The difference between distances to the foci always equals 2a.
Center at the origin
Foci $(\pm c,0)$ · vertices $(\pm a,0)$
Center $(0,0)$ · transverse axis along $y=0$
Vertices are at distance $a$ from center, and $c^2=a^2+b^2$ ($c>a$). Distance difference: $|PF_1-PF_2|=2a$.
Derive the equation from distance difference
Use a right triangle to find c
The corner of the guide rectangle is at (a, b). Its distance from the center is the focal distance c.
$c$ is the hypotenuse of a right triangle with legs $a$ and $b$.
For $a=3$ and $b=4$, $c=\sqrt{9+16}=\sqrt{25}=5$.
Dashed orange arc: radius c connects corner (a, b) to focus F.
Find a hyperbola’s vertices and asymptotes
The positive squared term gives the opening direction. Use a rectangle to find the asymptote slopes.
Horizontal: vertices (±3, 0); asymptotes y = ±(4/3)x.
- Mark the center and the vertices.
- Draw a rectangle centered at the origin: 3 units left and right, and 4 units up and down. For the vertical example, swap 3 and 4.
- Extend the rectangle’s diagonals to draw the asymptotes.
- Draw one branch from each vertex, approaching the asymptotes as it moves away from the center.
Rectangle corners are not points on the curve.
Shift a conic horizontally and vertically
Replacing x with x − h and y with y − k moves every point by h horizontally and k vertically.
- Center: $(h,k)$
- Vertices: $(h\pm5,k)$
- Foci: $(h\pm4,k)$
- Minor-axis endpoints: $(h,k\pm3)$
For a shifted parabola, $(x-h)^2=4p(y-k)$: $(h,k)$ is the vertex, the focus is $(h,k+p)$, and the directrix is $y=k-p$.
Exercises: three conic sections
For each conic: complete the square into standard form, find all key geometric features, and sketch the curve.
Exercise 1: Parabola
Exercise 2: Ellipse
Exercise 3: Hyperbola
Exercise: a parabola
Complete the square to find the vertex, opening direction, focus, and directrix.
1. Isolate the squared variable:
2. Complete the square on $x$:
3. Factor into standard form $(x-h)^2=4p(y-k)$:
- Vertex: $(h,k)=(-2,2)$
- $4p=-8\implies p=-2<0$ (opens downward)
- Focus: $(h,k+p)=(-2,2-2)=(-2,0)$
- Directrix: $y=k-p=2-(-2)\implies y=4$
- Axis of symmetry: $x=-2$
Exercise: an ellipse
Complete both squares to find the center, orientation, vertices, foci, and minor-axis endpoints.
1. Group terms and factor coefficients:
2. Complete both squares and balance the RHS:
3. Divide by 36 to get standard form:
- Center: $(1,-2)$ · horizontal major axis ($9>4\implies a=3, b=2$)
- $c=\sqrt{a^2-b^2}=\sqrt{9-4}=\sqrt{5}\approx2.24$
- Vertices: $(1\pm3,-2)\implies(-2,-2)\text{ and }(4,-2)$
- Foci: $(1\pm\sqrt{5},-2)$
- Minor-axis endpoints: $(1,-2\pm2)\implies(1,0)\text{ and }(1,-4)$
Exercise: a hyperbola
Watch the negative factor sign, determine the transverse axis, build the guide box, and find the asymptotes.
1. Group & factor — watch the minus sign:
Factoring $-9$ from $-36x$ leaves $+4x$.
2. Complete squares and balance the RHS:
Adding $4$ inside $-9(\dots)$ subtracts $36$, so subtract $36$ from RHS.
3. Standard form:
- Center: $(-2,1)$ · positive $y\implies$ vertical transverse axis
- $a=3$ (vert), $b=2$ (horiz) · $c=\sqrt{9+4}=\sqrt{13}\approx3.61$
- Vertices: $(-2,1\pm3)\implies(-2,4)\text{ and }(-2,-2)$
- Foci: $(-2,1\pm\sqrt{13})$
- Asymptotes: slopes $\pm\frac{a}{b}=\pm\frac32\implies\boxed{y-1=\pm\frac32(x+2)}$
Conic equations and key features
Complete the square → write standard form → find key points → sketch.
| Conic | Standard form (one orientation) | Key points |
|---|---|---|
| Parabola | $(x-h)^2=4p(y-k)$ | Focus $(h,k+p)$ Directrix $y=k-p$ |
| Ellipse | $\dfrac{(x-h)^2}{a^2}+\dfrac{(y-k)^2}{b^2}=1$ $a>b>0$; horizontal major axis | $c^2=a^2-b^2$ Distance sum $=2a$ |
| Hyperbola | $\dfrac{(x-h)^2}{a^2}-\dfrac{(y-k)^2}{b^2}=1$ | $c^2=a^2+b^2$ $y-k=\pm\dfrac ba(x-h)$ |