Ellipse Calculator
Enter the ellipse equation or the known data to get its equations (standard and general form), its elements (center, foci, vertices, semi-axes, latus rectum, eccentricity, area, etc.), and its graph.
x^2
x^\square
\frac{\square}{\square}
(
)
+
-
\times
=
Calculate ellipse
Clear
Ellipse equations
Standard form
General form
Ellipse elements
Orientation:
Center:
Foci
Major vertices
Minor vertices (co-vertices)
Semi-major axis:
Semi-minor axis:
Focal distance:
Latus rectum:
Eccentricity:
Axes of symmetry:
Area:
Perimeter:
x-intercepts
y-intercepts
Quick Examples
Center at origin
\dfrac{x^2}{25} + \dfrac{y^2}{9} = 1
Center not at origin
\dfrac{(x+3)^2}{16}+\dfrac{(y-4)^2}{36}=1
General form
4x^2+9y^2-16x-32=0
Center and axes
C(2, -1), 2a = 10, 2b = 6
Foci and point
F(±3, 0), P(0, 4)
Foci and vertex
F1 (1, -2), F2 (1, 4), V1 (1, 6)
Foci and major axis
F1 (-4, 3), F2 (2, 3), 2a = 10
Foci and minor axis
F(± 0, 3), 2b = 8
Four points
(3, 0), (-3, 0), (0, 2), (0, -2)
Rate this tool
Change my vote
Solved Exercises
Determine the general form and the elements of the ellipse \( \dfrac{x^2}{25} + \dfrac{y^2}{9} = 1. \)
Ellipse equations
Equation in standard form
$$ \dfrac{x^2}{25} + \dfrac{y^2}{9} = 1 $$
Equation in general form
$$ 9x^2 + 25y^2 - 225 = 0 $$
Ellipse elements
Orientation: Horizontal (major axis parallel to the x-axis).
Center: \( C \left(0, 0\right) \)
Foci
\( \begin{array}{l} F_1 \left(-4, 0\right) \\ \\ F_2 \left(4, 0\right)\end{array} \)
Major vertices
\( \begin{array}{l} V_1 \left(-5, 0\right) \\ \\ V_2 \left(5, 0\right) \end{array} \)
Minor vertices (co-vertices)
\( \begin{array}{l} V_3 \left(0, -3\right) \\ \\ V_4 \left(0, 3\right) \end{array} \)
Semi-major axis: \( a = 5 \)
Semi-minor axis: \( b = 3 \)
Focal distance: \( c = 4 \)
Latus rectum: \( L_R = \dfrac{18}{5} = 3.6 \)
Eccentricity: \( e = \dfrac{4}{5} = 0.8 \)
Axes of symmetry: \( x = 0, \quad y = 0 \)
Area: \( A = 15 \pi \approx 47.12 \)
Perimeter: \( P \approx 25.53 \)
x-intercepts
\( x_1 = -5 \\ \\ x_2 = 5 \)
y-intercepts
\( y_1 = -3 \\ \\ y_2 = 3 \)
Graph of the ellipse Calculate the elements of the ellipse with center not at the origin \( \dfrac{(x+3)^2}{16}+\dfrac{(y-4)^2}{36}=1. \)
Equations
Standard form
$$ \dfrac{\left(x + 3\right)^2}{16} + \dfrac{\left(y - 4\right)^2}{36} = 1 $$
General form
$$ 9x^2 + 4y^2 + 54x - 32y + 1 = 0 $$
Ellipse elements
Orientation: Vertical (major axis parallel to the y-axis).
Center: \( C \left(-3, 4\right) \)
Foci
\( \begin{array}{l} F_1 \left(-3, -2 \sqrt{5}+4\right) \approx \left(-3, -0.47\right) \\ \\ F_2 \left(-3, 2 \sqrt{5}+4\right) \approx \left(-3, 8.47\right)\end{array} \)
Major vertices
\( \begin{array}{l} V_1 \left(-3, -2\right) \\ \\ V_2 \left(-3, 10\right) \end{array} \)
Minor vertices (co-vertices)
\( \begin{array}{l} V_3 \left(-7, 4\right) \\ \\ V_4 \left(1, 4\right) \end{array} \)
Semi-major axis: \( a = 6 \)
Semi-minor axis: \( b = 4 \)
Focal distance: \( c = 2 \sqrt{5} \approx 4.47 \)
Latus rectum length: \( L_R = \dfrac{16}{3} \approx 5.33 \)
Eccentricity: \( e = \dfrac{\sqrt{5}}{3} \approx 0.75 \)
Axes of symmetry: \( x = -3, \quad y = 4 \)
Area of the ellipse: \( A = 24 \pi \approx 75.40 \)
Perimeter: \( P \approx 31.73 \)
x-intercepts
\( x_1 = 8\left(-\dfrac{\sqrt{5}}{6}-\dfrac{3}{8}\right) \approx -5.98 \\ \\ x_2 = 8\left(\dfrac{\sqrt{5}}{6}-\dfrac{3}{8}\right) \approx -0.02 \)
y-intercepts
\( y_1 = 18\left(-\dfrac{\sqrt{7}}{12}+\dfrac{2}{9}\right) \approx 0.03 \\ \\ y_2 = 18\left(\dfrac{\sqrt{7}}{12}+\dfrac{2}{9}\right) \approx 7.97 \)
Graph of the curve Obtain the elements and the standard form of the ellipse \( 4x^2+9y^2-16x-32=0. \)
Ellipse equations
Equation in standard form
$$ \dfrac{\left(x - 2\right)^2}{12} + \dfrac{y^2}{16/3} = 1 $$
Equation in general form
$$ 4x^2 + 9y^2 - 16x - 32 = 0 $$
Elements
Orientation: Horizontal (major axis parallel to the x-axis).
Center: \( C \left(2, 0\right) \)
Foci
\( \begin{array}{l} F_1 \left(-\dfrac{2 \sqrt{5}}{\sqrt{3}}+2, 0\right) \approx \left(-0.58, 0\right) \\ \\ F_2 \left(\dfrac{2 \sqrt{5}}{\sqrt{3}}+2, 0\right) \approx \left(4.58, 0\right)\end{array} \)
Major vertices
\( \begin{array}{l} V_1 \left(-2 \sqrt{3}+2, 0\right) \approx \left(-1.46, 0\right) \\ \\ V_2 \left(2 \sqrt{3}+2, 0\right) \approx \left(5.46, 0\right) \end{array} \)
Minor vertices (co-vertices)
\( \begin{array}{l} V_3 \left(2, -\dfrac{4}{\sqrt{3}}\right) \approx \left(2, -2.31\right) \\ \\ V_4 \left(2, \dfrac{4}{\sqrt{3}}\right) \approx \left(2, 2.31\right) \end{array} \)
Semi-major axis: \( a = 2 \sqrt{3} \approx 3.46 \)
Semi-minor axis: \( b = \dfrac{4}{\sqrt{3}} \approx 2.31 \)
Focal distance: \( c = \dfrac{2 \sqrt{5}}{\sqrt{3}} \approx 2.58 \)
Latus rectum: \( L_R = \dfrac{16}{3 \sqrt{3}} \approx 3.08 \)
Eccentricity: \( e = \dfrac{\sqrt{5}}{3} \approx 0.75 \)
Axes of symmetry: \( x = 2, \quad y = 0 \)
Area: \( A = 8 \pi \approx 25.13 \)
Perimeter: \( P \approx 18.32 \)
x-intercepts
\( x_1 = \dfrac{-16 \sqrt{3}+16}{8} \approx -1.46 \\ \\ x_2 = \dfrac{16 \sqrt{3}+16}{8} \approx 5.46 \)
y-intercepts
\( y_1 = -\dfrac{4 \sqrt{2}}{3} \approx -1.89 \\ \\ y_2 = \dfrac{4 \sqrt{2}}{3} \approx 1.89 \)
Graph of the ellipse on the plane Find the equations and elements of the ellipse with center at C(2, -1) , major axis 10 and minor axis 6.
Ellipse equations
Standard form
$$ \dfrac{\left(x - 2\right)^2}{25} + \dfrac{\left(y + 1\right)^2}{9} = 1 $$
General equation
$$ 9x^2 + 25y^2 - 36x + 50y - 164 = 0 $$
Elements
Orientation: Horizontal (major axis parallel to the x-axis).
Center: \( C \left(2, -1\right) \)
Foci
\( \begin{array}{l} F_1 \left(-2, -1\right) \\ \\ F_2 \left(6, -1\right)\end{array} \)
Major vertices
\( \begin{array}{l} V_1 \left(-3, -1\right) \\ \\ V_2 \left(7, -1\right) \end{array} \)
Minor vertices (co-vertices)
\( \begin{array}{l} V_3 \left(2, -4\right) \\ \\ V_4 \left(2, 2\right) \end{array} \)
Semi-major axis: \( a = 5 \)
Semi-minor axis: \( b = 3 \)
Focal distance: \( c = 4 \)
Latus rectum: \( L_R = \dfrac{18}{5} = 3.6 \)
Eccentricity: \( e = \dfrac{4}{5} = 0.8 \)
Axes of symmetry: \( x = 2, \quad y = -1 \)
Area: \( A = 15 \pi \approx 47.12 \)
Perimeter: \( P \approx 25.53 \)
x-intercepts
\( x_1 = \dfrac{25\left(-\dfrac{4 \sqrt{2}}{15}+\dfrac{4}{25}\right)}{2} \approx -2.71 \\ \\ x_2 = \dfrac{25\left(\dfrac{4 \sqrt{2}}{15}+\dfrac{4}{25}\right)}{2} \approx 6.71 \)
y-intercepts
\( y_1 = \dfrac{9\left(-\dfrac{2 \sqrt{7}}{5 \sqrt{3}}-\dfrac{2}{9}\right)}{2} \approx -3.75 \\ \\ y_2 = \dfrac{9\left(\dfrac{2 \sqrt{7}}{5 \sqrt{3}}-\dfrac{2}{9}\right)}{2} \approx 1.75 \)
Graph of the ellipse on the Cartesian plane Calculate the equations (standard and general form) and elements of the ellipse with foci at (±3, 0) and point (0, 4) .
Equations
Equation in standard form
$$ \dfrac{x^2}{25} + \dfrac{y^2}{16} = 1 $$
Equation in general form
$$ 16x^2 + 25y^2 - 400 = 0 $$
Elements
Orientation: Horizontal (major axis parallel to the x-axis).
Center: \( C \left(0, 0\right) \)
Foci
\( \begin{array}{l} F_1 \left(-3, 0\right) \\ \\ F_2 \left(3, 0\right)\end{array} \)
Major vertices
\( \begin{array}{l} V_1 \left(-5, 0\right) \\ \\ V_2 \left(5, 0\right) \end{array} \)
Minor vertices (co-vertices)
\( \begin{array}{l} V_3 \left(0, -4\right) \\ \\ V_4 \left(0, 4\right) \end{array} \)
Semi-major axis: \( a = 5 \)
Semi-minor axis: \( b = 4 \)
Focal distance: \( c = 3 \)
Latus rectum: \( L_R = \dfrac{32}{5} = 6.4 \)
Eccentricity: \( e = \dfrac{3}{5} = 0.6 \)
Axes of symmetry: \( x = 0, \quad y = 0 \)
Area: \( A = 20 \pi \approx 62.83 \)
Perimeter: \( P \approx 28.36 \)
x-intercepts
\( x_1 = -5 \\ \\ x_2 = 5 \)
y-intercepts
\( y_1 = -4 \\ \\ y_2 = 4 \)
Graph of the ellipse Determine the equations and elements of the ellipse with foci at (1, -2) and (1, 4) , and vertex at (1, 6) .
Ellipse equations
Standard form
$$ \dfrac{\left(x - 1\right)^2}{16} + \dfrac{\left(y - 1\right)^2}{25} = 1 $$
General form
$$ 25x^2 + 16y^2 - 50x - 32y - 359 = 0 $$
Ellipse elements
Orientation: Vertical (major axis parallel to the y-axis).
Center: \( C \left(1, 1\right) \)
Foci
\( \begin{array}{l} F_1 \left(1, -2\right) \\ \\ F_2 \left(1, 4\right)\end{array} \)
Major vertices
\( \begin{array}{l} V_1 \left(1, -4\right) \\ \\ V_2 \left(1, 6\right) \end{array} \)
Minor vertices (co-vertices)
\( \begin{array}{l} V_3 \left(-3, 1\right) \\ \\ V_4 \left(5, 1\right) \end{array} \)
Semi-major axis: \( a = 5 \)
Semi-minor axis: \( b = 4 \)
Focal distance: \( c = 3 \)
Latus rectum: \( L_R = \dfrac{32}{5} = 6.4 \)
Eccentricity: \( e = \dfrac{3}{5} = 0.6 \)
Axes of symmetry: \( x = 1, \quad y = 1 \)
Area: \( A = 20 \pi \approx 62.83 \)
Perimeter: \( P \approx 28.36 \)
x-intercepts
\( x_1 = 8\left(-\dfrac{\sqrt{2} \sqrt{3}}{5}+\dfrac{1}{8}\right) \approx -2.92 \\ \\ x_2 = 8\left(\dfrac{\sqrt{2} \sqrt{3}}{5}+\dfrac{1}{8}\right) \approx 4.92 \)
y-intercepts
\( y_1 = \dfrac{25\left(-\dfrac{\sqrt{3}}{2 \sqrt{5}}+\dfrac{2}{25}\right)}{2} \approx -3.84 \\ \\ y_2 = \dfrac{25\left(\dfrac{\sqrt{3}}{2 \sqrt{5}}+\dfrac{2}{25}\right)}{2} \approx 5.84 \)
Graph of the ellipse Related Tools
Leave a Reply