Ellipse
Polar form when the left focus point is at the origin:
Ellipse equation
x2 + y2 = 1
2 2
x2 + y2 =
Semi-major axis (a)
Semi-minor axis (b)
Area (A)
Accentricity (e)
Foci distance (c) ±
Flattening factor (g)
vertices ±
Ellipse center
Circumference (P)
Input limit:
NOTE: The perimeter of the ellipse is calculated by using accurate solution of infinite series to the accuracy selected. The increase of accuracy or the ratio a/b causes the calculator to use more terms to reach the selected accuracy.
 
The general ellipse equation with center not at (0,0) is:
x2 + y2 + x + y + = 0
(x )2  +  ( y )2 = 1
2 2
           
Ellipse Print ellipse
Ellipse
An ellipse is the locus of all points that the sum of whose distances from two fixed points is constant,
d1 + d2 = constant = 2a
the two fixed points are called the foci (or in single focus).
distance between both foci is:   2c
a and b − major and minor radius
Equation of an ellipse: Ellipse (center at   x = 0   y = 0)
The eccentricity  e  of an ellipse: Eccentricity
Where   (c = half distance between foci)         c < a         0 < e < 1
If e = 0 then the ellipse is a circle.
If 0 < e < 1 then it is an ellipse.
If e = 1 then the ellipse is a parabola.
If e > 1 then the ellipse is a hyperbola.
Ellipse
If   b > a   then the ellipse is a vertical ellipse and the foci are on the  y  axis.
In this case the equation of the ellipse is: Vertical ellipse
Example: Given the ellipse with equation    81x2 + 4y2 = 324
find ellipse parameters.
Solution: Divide by 324. to obtain Ellipse
Since   a < b   ellipse is vertical with foci at the   y   axis and   a = 9   and   b = 2.
Eccentricity: Eccentricity Flattening: Flattening
Focus c: c = a * e = 8.775 Area: A = π2∙9 = 56.55
The slop of the tangent line to the ellipse at point (x1 , y1) is: Slope of tangent
The tangent line equation at a point   (x1 , y1)   on the ellipse
Tangent line equation or Tangent line equation
The area of an ellipse: Ellipse area
From ellipse equation:
If the center of the ellipse is moved by     x = h   and   y = k   then the equations of the ellips become:
Ellipse center is at (h, k): Ellipse center
Ellipse center
If a point   y1   is given then: x1 value
If a point   x1   is given then: y1 value
Converting ellipse presentation formats:
Ax2 + By2 + Cx + Dy + E = 0 Ellipse general equation
① → ② Define: Define values
Define values
② → ① A = b2 B = a2 C = − 2hb2 D = − 2ka2 E = a2k2 + b2h2 − a2b2
Any point from the center to the circumference of the ellipse can be expressed by the angle θ   in the
range (0 − 2π)   as: x = a cosθ               y = b sinθ
If we substitute the values   x = r cosθ   and   y = r sinθ   in the equation of the ellipse we can get the
distance of a point from the center of the ellipse r(θ) as: Ellipse radius
If the origin is at the left focus then the ellipse equstion is:
Ellipse radius
The perimeter   (P)   of an ellipse is found by integration:
Perimeter of an ellipse
The only solution is by series: Perimeter of an ellipse
Perimeter of an ellipse
Where   e   is the eccentricity of the ellipse Perimeter of an ellipse
Another solution is by using the series: Perimeter of an ellipse
Perimeter of an ellipse
Where Perimeter of an ellipse
Ramanujan approximation for the circumference: Perimeter approximation
Where Perimeter approximation
Less accurate approximation Perimeter approximation
Vertical ellipse
From ellipse definition   d1 + d2 = 2a
From the drawing   d1 = d2
2d1 = 2a
And we get the relation   d1 = a   and:
Perimeter approximation
Question - verify the equation of an ellipse
Ellipse definition
From the definition of the ellipse we know that     d1 + d2 = 2a
Where  a  is equal to the x axis value or half the major axis.
From the drawing d1 and d2 are equal to:
Perimeter approximation
Perimeter approximation
Simplify the equation by transferring one redical to the right and squaring both sides:
Perimeter approximation
After rearranging terms we obtain: Perimeter approximation
We square again both sides to find: Perimeter approximation
After arranging terms we get: Perimeter approximation
Dividing by   a2 − c2   we find: Perimeter approximation
Since   a > c   we can introduce a new quantity: Perimeter approximation
And the equation of an ellipse is revealed: Perimeter approximation
Ellipse definition
If the foci are placed on the  y  axis then we can find the equation of the ellipse the same way:   d1 + d2 = 2a
Where  a  is equal to the y axis value or half the vertical axis.
From the drawing d1 and d2 are equal to:
Vertical ellipse constant distance
Vertical ellipse constant distance
After arranging terms and squaring we get: Vertical ellipse
After rearranging terms we find: Vertical ellipse
Set new quantity (see above): Vertical ellipse
Dividing by b2 we get the final form: Vertical ellipse equation
Exampe - Tangent line to an ellipse
Find the equation of the line tangent to the ellipse   4x2 + 12y2 = 1   at the point   P(0.25 , 0.25).
By implicit differentiation we will find the value of   dy/dx   that is the slope at any  x and y  point.
Implicit differentiation   dy/dx   is: Implicit differentiation
The value of dy/dx is: Vertical ellipse
At the given point the slope is: Ellipse tangent slope
Equation of the tangent line that passes through the point P and has slope m is:       y = mx + ( yp − mxp)
Substitute the point P(0.25 , 0.25) we get: y = mx + (0.25 - m * 0.25)
And the tangent line equation is: x + 3y − 1 = 0
Example - vertices and eccentricity
Find the equation of the ellipse that has vertices at (0 , ± 10) and has eccentricity of 0.8.
Notice that the vertices are on the  y  axis so the ellipse is a vertical ellipse and we have to use the vertical ellipse equation.
The equation of the eccentricity is: Implicit differentiation
After multipling by a we get: e2a2 = a2 − b2
The value of   b2   is: b2 = a2(1 − e2)
The equation of a vertical ellipse is: Vertical ellipse equation
And the final equation of the ellipse is: Vertical ellipse equation
Example - foci and eccentricity
Find the equation of the ellipse that has accentricity of 0.75, and the foci along 1. x axis 2. y axis, ellipse center is at the origin, and passing through the point (6 , 4).
The point (6 , 4) is on the ellipse therefore fulfills the ellipse equation.
1. Substitute the point (x1 , y1) into the ellipse
equation (foci at x axis):
Ellipse defined by a point
From the previous example: b2 = a2(1 − e2)
Substitute   b2   into ellipse equation: Vertical ellipse equation
The value of   a2   is: Vertical ellipse equation
The value of   b2   is: Vertical ellipse equation
And the equation of the ellipse is: 7x2 + 16y2 = 508
2. Vertical ellipse equation is (foci at y axis): Ellipse defined by a point
Substitute   b2   into ellipse equation: Vertical ellipse equation
The value of   a2   is: Vertical ellipse equation
The value of   b2   is: Vertical ellipse equation
And the equation of the ellipse is: 16x2 + 7y2 = 688
Example - transelated center of ellipse
Find the vertices and the foci coordinate of the ellipse given by     3x2 + 4y2 - 12x + 8y + 4 = 0.
Find the square in x and y: 3(x2 - 4x       ) + 4(y2 + 2y       ) = − 4
Add and subtruct 4 to the left parentheses and 1 to the right parentheses to obtain:
3(x − 2)2 − 12 + 4(y + 1)2 − 4 = − 4
3(x − 2)2 + 4(y + 1)2 = 12
After dividing by 12 we get: Ellipse equation
The center of this ellipse is at (2 , − 1)     h = 2   and   k = − 1.
Translate the ellipse axes so that the center will be at (0 , 0) by defining: x' = x − 2 y' = y + 1
now the ellipse equation in the x'y' system is: Ellipse equation
Which we recognize as an ellipse with vertices   a = ± 2   b vertixs   and the foci is   Foci distance
In the xy system we have the vertices at   (2 ± 2 , − 1) and the foci at   (2 ± 1 , − 1).
The sketch of the ellipse is: Ellipse equation
Example - distances from foci
Find the equation of the locus of all points the sum of whose distances from   (3, 0)   and   (9, 0)   is  12.
It can be seen that the foci are lying on the line   y = 0   so the ellipse is horizontal.
The focus is equal to: Foci distance
From the definition of the ellipse we know that: d1 + d2 = 2a
and the value of a is     12 = 2a a = 6
and the value of minor vertex   b   is: Foci distance
The ellipse in the   x'y'   system is: Foci distance
The ellipse in the   xy   system is: Foci distance
The ellipse after rearanging terms is: 3x2 + 4y2 − 36x = 0