Tangent Line to a Conic Section

To find the tangent line to a conic at a given point, use implicit differentiation.

For $x^{2}/25 + y^{2}/9 = 1$, differentiating both sides: $2x/25 + 2y \cdot y'/9 = 0$, so $y' = -9x/(25y)$.

At $(4, 9/5)$: $y' = -9(4)/(25 \cdot 9/5) = -36/45 = -4/5$.

Tangent line: $y - 9/5 = (-4/5)(x - 4)$, which simplifies to $y = -4x/5 + 25/5 = -4x/5 + 5$.

Verification: The tangent formula for an ellipse at point $(x_{0}, y_{0})$ is $xx_{0}/a^{2} + yy_{0}/b^{2} = 1$, giving $4x/25 + (9/5)y/9 = 1$, or $4x/25 + y/5 = 1$, or $y = 5 - 4x/5$. Same answer. ✓

Try it in the visualization

Drag the point along the ellipse and watch the tangent line update in real time. Toggle between ellipse, circle, parabola, and hyperbola to see tangent lines on each conic type. The slope value and tangent equation are computed live.