Hyperbola and Line Intersection Calculator

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

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

Formula & method

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

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

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

Multiplying by \( a^2 b^2 \) and expanding yields the quadratic \( p x^2 + q x + r = 0 \):

\[ p = b^2 - a^2 m^2 \] \[ q = -2hb^2 - 2a^2m(B - k) \] \[ r = b^2 h^2 - a^2(B - k)^2 - a^2 b^2 \]

The discriminant \( \Delta = q^2 - 4pr \) determines if there are 2, 1 (tangent), or 0 intersection points.

Hyperbola Parameters (Horizontal Axis)

\(h\) (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.