This calculator finds the points of intersection of an ellipse and a line.
The equation of the ellipse is
\[ \frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1 \]and the equation of the line (slope–intercept form) is
\[ y=mx+B \]Substituting \( y=mx+B \) into the ellipse equation gives
\[ \frac{(x-h)^2}{a^2}+\frac{(mx+B-k)^2}{b^2}=1 \]This simplifies to a quadratic equation in \(x\):
\[ (b^2+a^2m^2)x^2+(-2hb^2+2ma^2B-2ma^2k)x+ (b^2h^2+a^2k^2+a^2B^2-2a^2Bk-a^2b^2)=0 \]To find the points of intersection, the calculator solves this quadratic equation for the \(x\)-coordinates, then substitutes each value into
\[ y=mx+B \]to obtain the corresponding \(y\)-coordinates.
1. Enter the center \((h,k)\) of the ellipse and the constants \(a\) and \(b\).
2. Enter the slope \(m\) and \(y\)-intercept \(B\) of the line.
3. Click Calculate.
The coordinates of the intersection points \(P_1\) and \(P_2\) will be displayed.
Note: There may be two intersection points, one (tangent case), or none.