 # Find Points Of Intersection of Ellipse and Line - Calculator

A calculator to find the points of intersection of a ellipse and a line.

## Intersection of Ellipse and Line

The equation of the ellipse is of the form
(x - h)2 / a2 + (y - k)2 / b2 = 1
and the equation of the line is of the slope intercept form
y = m x + B
If y = m x + B is substituted into
(x - h)2 / a2 + (y - k)2 / b2 = 1
we end up with a quadratic equation given by:
(x - h)
2 / a2 + (m x + B - k)2 / b2 = 1
which may be rearranged as a quadratic equation given by
(b
2 + a2 m2) x2 + (-2 h b2 + 2 m a2 B - 2 m a2 k) x + (b2 h2 + a2 k2 + a2 B2 - 2 a2 B k - a2 b2) = 0
To find the points of intersection, this calculator solves the above equation to find the x coordinates and then uses equation y = m x + B to find the y coordinates.

## Use Calculator to Find Points Of Intersection of Ellipse and Line

1 - Enter the coordinates (h , k) of the center of the ellipse and the constant a and b then enter the slope m of the line and its y intercept B; then press "Calculate". The x and y coordinates of the two points of intersection P1 and P2 are displayed.
Note that this problem may have two points of intersection, one point of intersection or no points of intersection.

 Ellipse: (h , k) are the coordinates of the center of the ellipse and a and b are the length of the major and minor axes. h = 1 , k = 1 a = 1 , b = 1 Line m = 1 , B = 1 Decimal places = 4 Coordinates of the points of intersection P1( , ) P2( , )