If the center of the ellipse is moved by x = h and y = k then if a point on the ellipse is given the corresponding x or y coordinate is calculated by the equations:
Ellipse equation |
Given |
Equivalent point on ellipse |
![(x+h)^2/a^2 +(y+k)^2/b^2 =1](/TrigoCalc/Circles2/Ellipse/Eq/eq2.png) |
x1 |
![y_(1,2)=-k±b/a √(a^2-(x_1+h)^2 )](/TrigoCalc/Circles2/Ellipse/Eq/eq2c.png) |
y1 |
![x_1,2=-h±a/b √(b^2-〖y_1〗^2-2ky_1-k^2 )](/TrigoCalc/Circles2/Ellipse/Eq/eq2b.png) |
Notice: these equations are good for horizontal and vertical ellipses.
Ellipse |
![Horizontal ellipse](/TrigoCalc/Circles2/Ellipse/Images/Ellipse5.png) |
![Vertical ellipse](/TrigoCalc/Circles2/Ellipse/Images/Ellipse5a.png) |
Center |
(h , k) |
(h , k) |
Vertices |
(h − a , k) (h + a , k) |
(h , k − a) (h , k + a) |
Foci |
(h − c , k) (h + c , k) |
(h , k − c) (h , k + c) |
The slope of the line tangent to the ellipse at point (x1 , y1) is:
The equation of the tangent line at point (x1 , y1) on the ellipse is:
![(b^2 (x_1+h))/(a^2 (y_1+k) ) x+y-y_1-(b^2 (x_1+h))/(a^2 (y_1+k) ) x_1=0](/TrigoCalc/Circles2/Ellipse/Eq/eq2b1.png)
Or: |
![y=-(b^2 (x_1+h))/(a^2 (y_1+k) ) x+y_1+(b^2 (x_1+h))/(a^2 (y_1+k) ) x_1](/TrigoCalc/Circles2/Ellipse/Eq/eq2a.png) |
① |
Ax2 + By2 + Cx + Dy + E = 0 |
② |
![Ellipse general equation](/TrigoCalc/Circles2/Ellipse/Eq/eq16b.png) |
① → ② |
Define: |
![Define values](/TrigoCalc/Circles2/Ellipse/Eq/eq16.png) |
![Define values](/TrigoCalc/Circles2/Ellipse/Eq/eq16a.png) |
② → ① |
A = b2 |
B = a2 |
C = 2hb2 |
D = 2ka2 |
E = a2k2 + b2h2 − a2b2 |
Polar coordinate of ellipse:
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](/TrigoCalc/Circles2/Ellipse/Eq/eq2d.png) |
If the origin is at the left focus then the ellipse equation is:
|