Famous Theorems of Mathematics/Geometry/Conic Sections

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Parabola Properties[edit]

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

and that the vertex of this parabola is (h,k)

Statement Reason
(1) Arbitrary real value h Given
(2) Arbitrary real value k Given
(3) Arbitrary real value p where p is not equal to 0 Given
(4) Line l, which is represented by the equation Given
(5) Focus F, which is located at Given
(6) A parabola with directrix of line l and focus F Given
(7) Point on parabola located at Given
(8) Point (x, y) must is equidistant from point f and line l. Definition of parabola
(9) The distance from (x, y) to l is the length of line segment which is both perpendicular to l and has one endpoint on l and one endpoint on (x, y). Definition of the distance from a point to a line
(10) Because the slope of l is 0, it is a horizontal line. Definition of a horizontal line
(11) Any line perpendicular to l is vertical. If a line is perpendicular to a horizontal line, then it is vertical.
(12) All points contained in a line perpendicular to l have the same x-value. Definition of a vertical line
(13) Point has a y-value of . (4) and (9)
(14) Point has an x-value of x. (7), (9), and (12)
(15) Point is located at (x, k - p). (13) and (14)
(16) Point is located at (x, y). (9)
(17) Distance Formula
(18) Distributive Property
(19) Apply square root; distance is positive
(20) Distance Formula
(21) Distributive Property
(22) Definition of Parabola
(23) Substitution
(24) Square both sides
(25) Distributive property
(26) Subtraction Property of Equality
(27) Addition Property of Equality; Subtraction Property of Equality
(28) Distributive Property

Finding the Axis of Symmetry[edit]

Statement Reason
(29) The axis of symmetry is vertical. (10); Definition of axis of symmetry; if a line is perpendicular to a horizontal line, then it is vertical
(30) The axis of symmetry contains (h, k + p). Definition of Axis of Symmetry
(31) All points in the axis of symmetry have an x-value of h. Definition of a vertical line; (30)
(32) The equation for the axis of symmetry is . (31)

Finding the Vertex[edit]

Statement Reason
(33) The vertex lies on the axis of symmetry. Definition of the vertex of a parabola
(34) The x-value of the vertex is h. (33) and (32)
(35) The vertex is contained by the parabola. Definition of vertex
(36) (35); Substitution: (28) and (34)
(37) Simplify
(38) Division Property of Equality
(39) Addition Property of Equality
(40) Symmetrical Property of Equality
(41) The vertex is located at . (34) and (40)