Ellipse and Line Intersection Calculator

Find Intersection Points: Ellipse & Line with Step-by-Step Solutions

Solve the system of equations to find where an ellipse and a line meet. Complete step-by-step explanation shown for every calculation.

Formula & method

\[ \text{Ellipse: } \frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1 \] \[ \text{Line: } y = m x + B \]

Substituting \( y = mx + B \) into the ellipse equation gives:

\[ \frac{(x - h)^2}{a^2} + \frac{(mx + B - k)^2}{b^2} = 1 \]

Multiplying by \( a^2 b^2 \) and expanding yields the quadratic equation:

\[ (b^2 + a^2 m^2) x^2 + (-2hb^2 + 2m a^2 B - 2m a^2 k) x \] \[ + (b^2 h^2 + a^2 k^2 + a^2 B^2 - 2a^2 B k - a^2 b^2) = 0 \]

The discriminant determines if there are 2, 1 (tangent), or 0 intersection points.

Ellipse Parameters

\(h\) (x-coordinate of center)

\(k\) (y-coordinate of center)

\(a\) (semi-axis along x) *

\(b\) (semi-axis along y) *

Line Parameters

\(m\) (slope)

\(B\) (y-intercept)

Decimal places

Intersection Points

Enter parameters and press "Find Intersection (Show Steps)"

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Step-by-step solution will appear here after calculation.