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单词 TangentOfConicSection
释义

tangent of conic section


The equation of every conic sectionMathworldPlanetmath (and the degenerate cases) in the rectangular(x,y)-coordinate systemMathworldPlanetmath may be written in the form

Ax2+By2+2Cxy+2Dx+2Ey+F=0,

where A, B, C, D, E and F are constants and  A2+B2+C2>0.11This is true also in any skew-angled coordinate system.   (The 2Cxy is present only if the axes are not parallelMathworldPlanetmathPlanetmath to the coordinate axes.)

The equation of the tangent line of an ordinary conic section (i.e., circle, ellipseMathworldPlanetmath, hyperbolaMathworldPlanetmath and parabola) in the point (x0,y0) of the curve is

Ax0x+By0y+C(y0x+x0y)+D(x+x0)+E(y+y0)+F=0.

Thus, the equation of the tangent line can be obtained from the equation of the curve by polarizing it, i.e. by replacing

x2 with x0x,  y2 with y0y,  2xy with y0x+x0y,  2x with x+x0,  2y with y+y0.

Examples:  The of the ellipse  x2a2+y2b2=1   is  x0xa2+y0yb2=1, the of the hyperbola  xy=12   is  y0x+x0y=1.

Titletangent of conic section
Canonical nameTangentOfConicSection
Date of creation2013-03-22 14:28:40
Last modified on2013-03-22 14:28:40
Ownerpahio (2872)
Last modified bypahio (2872)
Numerical id16
Authorpahio (2872)
Entry typeDefinition
Classificationmsc 51N20
Synonymtangent of quadratic curve
Related topicTangentLine
Related topicTangentOfCircle
Related topicTangentPlaneOfQuadraticSurface
Related topicQuadraticInequality
Related topicConjugateDiametersOfEllipse
Related topicConjugateHyperbola
Related topicQuadraticCurves
Related topicEquationOfTangentOfCircle
Related topicTangentOfHyperbola
Definespolarising
Definespolarizing
Definespolarize
Definesmixed term
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更新时间:2025/5/4 16:48:41