Conic section/Proofs

In mathematics, conic sections are relations which represent the equation of the curve (or curves) that result from passing a plane through a cone.

Circles

Definition: The locus of all points in a plane which are equidistant from a given point. This given point is known as the circle's center, and the set distance from the center is known as the radius, represented by the letter "r".

In other words, in a circle with a center (h, k), and a radius of "r", a point $(x,y)$ in the circle is "r" units away from the center. With this, one can insert these variables into the distance formula, which can be modeled by the equation:

: $d\; =\; sqrt\{(x\_1-x\_2)^2+(y\_1-y\_2)^2\}$

where 'd' is the distance between two points with coordinates $(x\_1,y\_1)$ and $(x\_2,y\_2)$. Because "r" is the distance between points $(h,k)$ and $(x,y)$, "r" can be substituted for "r". ("x", "y") can replace $(x\_1,y\_1)$ and ("h", "k") can replace $(x\_2,y\_2)$:

: $r\; =\; sqrt\{(x-h)^2+(y-k)^2\}.,$

By squaring both sides, one is left with the final equation:

: $r^2\; =\; (x-h)^2+(y-k)^2.,$

Parabolas

Definitions

*Directrix: line "l"
*Focus: point "f" which is not contained by line "l"
*Parabola: the locus of points in a plane which are equidistant from line "l" and a point "f"
*Axis of symmetry: the line which is both perpendicular to the directrix and contains point "f"
*vertex: the locus of points which lie on a the parabola and are points on the axis of symmetry

Proof

Prove that for point ("x", "y") on a parabola with vertex ("h","k"), focus ("h", "k" + "p"), and directrix "y" = "k" − "p":

: $(x\; -\; h)^2\; =\; 4p(y\; -\; k),$

Ellipses

For a great discussion of ellipses see the wikipedia article Ellipse