Tangent and Normal to a Conic Calculator
Tangents and chords of any conic from its equation: the tangent and normal at a point, the chord of contact (polar) and pair of tangents from an outside point, the chord with a given midpoint, and the co-normal points whose normals pass through a point.
About the Tangent and Normal to a Conic Calculator
Write the conic as S = ax² + 2hxy + by² + 2gx + 2fy + c = 0 and, for a point (x₁, y₁), form T by replacing x² with xx₁, y² with yy₁, 2xy with xy₁ + yx₁, 2x with x + x₁ and 2y with y + y₁. S₁ is S at the point.
T = 0 is the tangent at a point on the conic, and the chord of contact (the polar) from a point off it. T = S₁ is the chord bisected at the point, and SS₁ = T² is the pair of tangents from it.
The formula
Tangent / chord of contact: T = 0; chord with midpoint: T = S₁; pair of tangents: SS₁ = T²
Worked example
x²/16 + y²/9 = 1 at (16/5, 9/5)
- T: (16/5)x/16 + (9/5)y/9 − 1 = 0
- x/5 + y/5 = 1, so x + y = 5
Things to keep in mind
- Coefficients must be fractions. With a surd in the point the lines are given in decimals.
Questions people ask
What is a polar?
For any point P, the polar of P is the line T = 0. When P is outside the conic it is the chord of contact.
What are co-normal points?
Points on a conic whose normals pass through one point; a parabola has up to three and an ellipse up to four. Choose 'Normals through a point' to find them.