Conic Sections Calculator

Result
y^2 = 8x

Vertex \displaystyle \left(0,\ 0\right), opening to the right.

  • Type: Parabola
  • Method: Complete the square
  • Vertex: \displaystyle \left(0,\ 0\right)
  • Focus: \displaystyle \left(2,\ 0\right)
  • Directrix: \displaystyle x = -2
  • Axis of symmetry: \displaystyle y = 0
  • Opens: to the right

Step-by-step solution

The problem

y^{2} = 8x
1

Identify the conic

Only \displaystyle y is squared, so it is a parabola that opens left or right.

2

Vertex, focus and directrix

\begin{aligned} \text{vertex } \left(0,\ 0\right),\quad p &= 2 \\[4pt] \text{focus } \left(2,\ 0\right),\quad \text{directrix } x &= -2 \end{aligned}

Compare with \displaystyle (y - h)^2 = 4p(x - k): \displaystyle 4p = 8, so \displaystyle p = 2. The focus is \displaystyle p from the vertex inside the curve; the directrix is \displaystyle p on the other side.

Graph

  • \displaystyle y^{2} = 8x

The parabola with its key points.

Vertex
(0, 0)
Focus
(2, 0)
−8−6−4−2246810−4−224(0, 0) vertex(2, 0) focus

Type an equation with x and y, like 9x^2 + 4y^2 - 18x + 16y - 11 = 0. For an upright parabola start with parabola: parabola y = x^2 - 4x + 3.

Try an example

Circle x^2 + y^2 - 4x + 6y - 12 = 0x^2 + y^2 = 25
Ellipse 4x^2 + 9y^2 = 369x^2 + 4y^2 - 18x + 16y - 11 = 0
Hyperbola x^2/16 - y^2/9 = 1y^2 - 4x^2 + 8x - 13 = 0
Parabola y^2 = 8xparabola y = x^2 - 4x + 3y^2 + 4y - 8x + 12 = 0

Paste the equation of a curve and the calculator names it – circle, ellipse, parabola or hyperbola – completes the square and lists its centre, vertices, foci, directrix or asymptotes, with a graph. x² + y² − 4x + 6y − 12 = 0, for instance, is a circle with centre (2, −3) and radius 5.

Which conic is it?

For Ax^2 + Cy^2 + Dx + Ey + F = 0 (no xy term), the coefficients of the squares decide:

SquaresConicStandard form
A = CCircle(x - h)^2 + (y - k)^2 = r^2
A \ne C, same signEllipse\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1
Opposite signsHyperbola\frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1
Only one is squaredParabola(x - h)^2 = 4p(y - k)

Completing the square in x and y

Example: x^2 + y^2 - 4x + 6y - 12 = 0

\begin{aligned} (x^2 - 4x) + (y^2 + 6y) &= 12 \\ (x^2 - 4x + 4) + (y^2 + 6y + 9) &= 12 + 4 + 9 \\ (x - 2)^2 + (y + 3)^2 &= 25 \end{aligned}

A circle with centre (2, -3) and radius 5.

An ellipse: 9x^2 + 4y^2 - 18x + 16y - 11 = 0. Take out the coefficients before completing the square:

9(x - 1)^2 + 4(y + 2)^2 = 11 + 9 + 16 = 36 \;\Rightarrow\; \frac{(x - 1)^2}{4} + \frac{(y + 2)^2}{9} = 1

Centre (1, −2). The bigger denominator, 9, is under y, so the long axis is vertical: a = 3, b = 2, c = \sqrt{9 - 4} = \sqrt{5}. Vertices (1, 1) and (1, −5); foci (1, -2 \pm \sqrt{5}). Open it in the calculator.

A hyperbola: x² − y² = 4 divides by 4 to \frac{x^2}{4} - \frac{y^2}{4} = 1 – it opens left and right, with vertices (±2, 0), foci (\pm 2\sqrt{2}, 0) and asymptotes y = ±x. A parabola: y² = 8x is y^2 = 4px with p = 2 – focus (2, 0), directrix x = −2, opening to the right.

Key features

  • Ellipse: the larger denominator is a^2; vertices are a from the centre along the major axis, foci c from the centre with c^2 = a^2 - b^2; eccentricity e = \frac{c}{a}.
  • Hyperbola: a^2 is under the positive term; c^2 = a^2 + b^2; asymptotes y - k = \pm\frac{b}{a}(x - h) for one opening left and right.
  • Parabola: vertex (h, k); the focus is p from the vertex inside the curve and the directrix is p on the other side.

Conics in the real world

  • Parabolas send every ray that arrives parallel to the axis through the focus – which is why satellite dishes, car headlights and solar cookers are parabolic, with the receiver or bulb at the focus.
  • Ellipses are the paths of planets, with the Sun at one focus. Earth’s orbit has eccentricity about 0.017 – almost a circle; Halley’s comet’s is about 0.97, a long thin ellipse.
  • Hyperbolas appear where two signals arrive with a fixed time difference – the idea behind older radio navigation systems – and as the shape of a fast comet’s path that passes the Sun once and never returns.

To solve a quadratic in one variable, use the quadratic equation solver.

Frequently asked questions

What are the foci of an ellipse?

Two points on the major axis, c from the centre, where c² = a² − b². The sum of the distances from any point on the ellipse to the two foci is always 2a.

Why does y = x² − 4x + 3 need the word “parabola”?

Without it the calculator treats the equation as something to solve for x. With “parabola” in front it finds the vertex, focus and directrix instead.

What if the equation has an xy term?

An xy term means the conic is rotated; that case is not handled here yet.