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_{1} 

y_{1} 

Notice: these equations are good for horizontal and vertical ellipses.
Ellipse 


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:
Or: 

① 
Ax^{2} + By^{2} + Cx + Dy + E = 0 
② 

① → ② 
Define: 


② → ① 
A = b^{2} 
B = a^{2} 
C = 2hb^{2} 
D = 2ka^{2} 
E = a^{2}k^{2} + b^{2}h^{2} − a^{2}b^{2} 
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: 

If the origin is at the left focus then the ellipse equation is:
