

tangent line
Definition/Summary

The tangent to a curve in a plane at a particular point has the same Gradient as the curve has at that point.
More generally, the (n1)dimensional tangent hyperplane to an (n1)dimensional surface in ndimensional space at a particular point has the same Gradient as the surface has at that point.
So if [itex]A\,=\,(a_1,a_2,\cdots a_n)[/itex] is a point on a surface defined by the equation [itex]F(x_1,x_2,\cdots x_n) = 0[/itex], then the tangent hyperplane to the curve through [itex]A[/itex] is [itex]\frac{\partial F}{\partial x_1}\arrowvert_A(x_1  a_1)\,+\,\frac{\partial F}{\partial x_2}\arrowvert_A(x_2  a_2)\,+\,\cdots\,\frac{\partial F}{\partial x_2}\arrowvert_A(x_n  a_n)\,=\,0[/itex]
If a curve in n dimensions is defined using a parameter t as [itex]A(t)\,=\,(a_1(t),a_2(t),\cdots a_n(t))[/itex] , then its tangent is:
[itex](x_1  a_1) / \frac{da_1}{dt}\,=\,(x_2  a_2) / \frac{da_2}{dt}\,=\,\cdots\,=\,(x_n  a_n) / \frac{da_n}{dt}[/itex] 
Equations

For example, if the point [itex]A\,=\,(x_0,y_0)[/itex] lies on the circle:
[tex]F(x,y)\,=\,(xp)^2+(yq)^2\,\,r^2\,=\,0[/tex] (1)
then [tex]\frac{\partial F}{\partial x}\arrowvert_A\,=\,2(xp)\arrowvert_A\,=\,2(x_0p)[/tex]
and [tex]\frac{\partial F}{\partial y}\arrowvert_A\,=\,2(yq)\arrowvert_A\,=\,2(y_0q)[/tex]
and so the equation of the tangent at [itex]A[/itex] is:
[tex](x_0p)(xx_0)\,+\,(y_0q)(yy_0)=0[/tex] (2)
Alternatively, the same circle can be defined by [itex]A(\theta)\,=\,(p+r\cos\theta, q+r\sin\theta)[/itex]
and so the equation of the tangent at [itex]A(\theta)[/itex] is:
[tex]\frac{xpr\cos\theta}{r\sin\theta}\,=\,\frac{yqr\sin\theta}{r\cos\theta}[/tex] 


Breakdown

Mathematics
> Geometry
>> Coordinate Geometry

Images



Extended explanation

We will transform the equation (2) into more convenient type for better way of memorizing and using the formula. Because of [itex]M(x_1,y_1) /in K[/itex]:
[tex](x_1p)+(y_1q)^2=r^2[/tex] (3)
If we sum the equations (2) and (3), we get:
[tex](x_1p)(xx_1)+(y_1q)(yy_1)+(x_1p)+(y_1q)^2=r^2[/tex]
[tex](x_1p)[xx_1+x_1p]+(y_1q)[yy_1+y_1q]=r^2[/tex]
[tex](x_1p)(xp)+(y_1q)(yq)=r^2[/tex] (4)
The equation (4) is equation of tangent of the circle in the point [itex]M(x_1,y_1) \in K[/itex].
If the K have center (0,0), i.e [itex]K: x^2+y^2=r^2[/itex], then p=q=0, so the equation of the tangent is:
[tex]x_1x+y_1y=r^2[/tex] 
Commentary
