[1] a = |AA'|/2 is called also the

[2] O is the middle of AA', the circle c centered at O, with radius a is called the

[3] Defining the

[4] Defining b properly, this is equivalent with the equation below, and related to the figure by: |HJ|/|XJ| = a/b, b is called the

[5] Defining the quantities e, h, through the equations of the first line below, the equation of the conic becomes equal to the equation of the second line, depicting another geometric property of the ellipse:

[6] Taking the line parallel to the y-axis at Q(h,0), called a

[7] There is a second directrix: the symmetric of QN w.r. to O, characterized by the same property as the previous one. The two directrices are the polars of the foci, with respect to the conic.

[8] The direction of the normal at X(x,y) is (x/a

[9] This implies that the symmetric F* of F with respect to the tangent XT is on XF' and the middle G of FF* is on the auxiliary circle with diameter AA'.

[10] Circle d, with center at the focus F' and radius 2a is called a

[11] Projecting a focal point on a tangent, defines a point on the auxiliary circle c (by [9] OG is parallel and the half length of F'F*).

[12] Line XJ is simultaneously the polar of T with respect to the ellipse and the auxiliary circle c. Therefore the tangents at X to the ellipse and H to the circle meet at T. Also lines KX and PH meet at the same point on the x-axis since JX/JH = OK/OP = b/a. The file CommonPolar.html contains some consequences of the coincidence of the tangent lines at T.

[13] All properties of ellipses, discussed so far, have their twins for hyperbolas (see Hyperbola.html ). Ellipses and hyperbolas are the two kinds of conics called collectively

Director.html

Hyperbola.html

Parabola.html

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