Ellipse line intersection calculator Print ellipse line calculator ▲
Ellipse equation
x2 + y2 + x + y + = 0
(x − )2  +  ( y − )2  = 1
2 2
Line equation of the form     mx + ny + c = 0
x + y + = 0
Intersection coordinates:
Notes and input limit
           
Ellipse and line intersection summary Print ellipse and line intersection summary ▲
Ellipse and line general equations
Ellipse general form
Line equation:       y = mx + c
 Ellipse
Intersection points as a function of  h, k  and  c  are:
If         h = 0,       k = 0,       c = 0
ellipse form: General form of an ellipse with center at (0 , 0)
Line form: y = m x
x_1,2=±ab/√(a^2 m^2+b^2 )
y_1,2=±abm/√(a^2 m^2+b^2 )Or y1,2 = m x1,2
If         h = 0,       k = 0,       c ≠ 0
ellipse form: General form of an ellipse with center at (0 , 0)
Line form: y = m x + c
x_1,2=(-a^2 mc±ab√(a^2 m^2+b^2-c^2 ))/(b^2+a^2 m^2 )
y_1,2=(b^2 c±abm√(a^2 m^2+b^2-c^2 ))/(b^2+a^2 m^2 )Or      y1,2 = m x1,2 + c
If         h ≠ 0,       k ≠ 0,       c ≠ 0
ellipse form: Ellipse general form
Line form: y = m x + c
x_(1,2)=(b^2 h-a^2 mφ±ab√(b^2+a^2 m^2-2mφh-φ^2-m^2 h^2 ))/(b^2+a^2 m^2 )
y_(1,2)=(b^2 δ+a^2 m^2 k±abm√(b^2+a^2 m^2+2δk-k^2-δ^2 ))/(b^2+a^2 m^2 )
Or             y1,2 = m x1,2 + c
Where:         φ = c − k         and         μ = c + m h
If         h ≠ 0,       k ≠ 0,       c ≠ 0,       x = d
ellipse form: (-h)^2/a^2 +(y-k)^2/b^2 =1
Line form: x = d
(vertical line)
y_(1,2)=k±b/a √(a^2-d^2-h^2+2dh)
x1,2 = d
If         h ≠ 0,       k ≠ 0,       c ≠ 0,       y = d
ellipse form: (x-h)^2/a^2 +(d-k)^2/b^2 =1
Line form: y = d
(horizontal line)
x_(1,2)=h±a/b √(b^2-d^2-k^2+2dk)
y1,2 = d
Verify the equations of an ellipse and line intersection Print ellipse and line intersection summary ▲
Find the intersection points of an ellipse and a line
Ellipse
The ellipse equation is given by:
Ellipse general form (1)
Line equation: y = mx + c (2)
The center of the ellipse is at  (h , k)
Substitute eq. (2) into eq. (1)
(x-h)^2/a^2 +(mx+c-k)^2/b^2 =1
Solving for x we have:
b2(x − h)2 + a2(mx + φ)2 − a2b2 = 0 where   φ = c − k
b2x2 − 2hb2x + b2h2 + a2m2x2 + 2a2mφx + φ2 − a2b2 = 0
x2(b2 + a2m2) + x(2a2mφ − 2hb2) + b2h2 + φ2 − a2b2 = 0
The solution of this quadratic equation is:
x_1,2=(b^2 h-a^2 mε±√(a^4 m^2 ε^2-2a^2 mεb^2 h+b^4 h^2-b^4 h^2-b^2 a^2 ε^2+b^4 a^2-a^2 m^2 b^2 h^2-a^4 m^2 ε^2+a^4 m^2 b^2 ))/(b^2+a^2 m^2 )
x_1,2=(b^2 h-a^2 mφ±√((a^2 mφ-b^2 h)^2-(b^2+a^2 m^2 )(b^2 h^2+a^2 φ^2-a^2 b^2 ) ))/(b^2+a^2 m^2 )
Finally, we get: x_(1,2)=(b^2 h-a^2 mφ±ab√(b^2+a^2 m^2-2mφh-φ^2-m^2 h^2 ))/(b^2+a^2 m^2 ) (3)
Note: for each  x  value there are two possible  y  coordinates but only one coordinate is on the ellipse see the red points in the drawing, if we find the numeric values of  x  and substitute it in the  y  equation we get automatically the correct  y  coordinate. If we use the general equation for  y  as found below we have to check the correct  y  coordinate for each  x.
To find the  y  intersection coordinates we substitutes x=(y-c)/m into equation  (1).
x=(y-c)/m
For ease of calculations, we denote         μ = c + m h
b2 (y − μ)2 + a2 m2 (y − k)2 − a2 m2 b2 = 0
b2 y2 − b2 2yμ + b2 μ22 + a2 m2 y2 − a2 m2 2yk + a2 m2 k2 − a2 m2 b2 = 0
y2 (b2 + a2 m2) − y(2b2 μ + 2a2 m2 k) + b2 μ2 + a2 m2 k2 − a2 m2 b2 = 0
Solving this quadratic equation and denote:       Φ = b2 μ2 + a2 m2 k2 − a2 m2 b2       we get:
x=(y-c)/m
Finally, we get: y_(1,2)=(b^2 δ+a^2 m^2 k±abm√(b^2+a^2 m^2+2δk-k^2-δ^2 ))/(b^2+a^2 m^2 ) (4)
We could simply substitute the values of   x1,2   into equation   (2)     y = mx + d     and get immediately the correct values of   y1   and   y2
Numeric example of intersection between ellipse and line Print ellipse and line intersection summary ▲
Find the coordinates of the intersection of an ellipse given by the equation     (x-2)^2/9+(y+3)^2/36=1
and the line given by    2x − 4y − 5 = 0
From the equation of the ellipse, we can see that   a < b   so the ellipse is a vertical ellipse with vertices at:
the y axis at: (h , k −b) → (2 , −3 − 6) → (2 , −9)
and (h , k + b) → (2 , −3 + 6) → (2 , 3)
We have             φ = c − k = −5/4 + 3 = 1.75            m = 0.5            a = 3   and   b = 6
Solving equation   (3)   for   x1,2   we get:
x_1,2=(36∙2-9∙0.5∙1.75±3∙6√(36+9∙0.25-2∙0.5∙1.75∙2-〖1.75〗^2-〖0.5〗^2∙4))/(36+9∙〖0.5〗^2 )
x_1,2=(64.125±18√30.6875)/38.25=(64.125±99.713)/38.25=4.28 ,-0.93
From the equation of the line, we can get the y coordinates: y=(2x-5)/4=0.5x-1.25
y1 = 0.5 · 4.28 − 1.25 = 0.89 y2 = 0.5 · (−0.93) − 1.25 = −1.72
And the intersection coordinates are:       (4.28 , 0.89)     and     (−0.93 , −1.72)
If we find the y coordinates according to equation (4) we get:
y_1,2=(36∙(-0.25)+9∙0.25∙(-3)±6∙3∙0.5√(36+9∙0.25+2∙(-0.25)(-3)-9-0.0625))/(36+9∙0.25)
y_1,2=(-15.75±9√30.6875)/38.25=(-15.75±49.86)/38.25=0.89 -1.72
Now we have to decide which  y  is the correct value for each  x  coordinate because the intersection point for example can be:   (4.28 , 0.89)   or   (4.28 , −1.72) for that reason the better method to solve  y  coordinate is by using the value of  y  in the line equation as described before.
Now we have to check which pair of intersection point is located on the ellipse contour.
Check point   (4.28 , −1.72) (x_1-2)^2/9+(y_1+3)^2/36=(4.28-2)^2/9+(-1.72+3)^2/36=0.623
Check point   (4.28 , 0.89) (x_1-2)^2/9+(y_2+3)^2/36=(4.28-2)^2/9+(0.89+3)^2/36=1
We clearly see that the correct intersection point is:   (4.28 , 0.89) and the second point is the remaining coordinates   (−0.93 , −1.72)