Parabola/Proofs

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This mathematics article is devoted entirely to providing mathematical proofs and support for claims and statements made in the article parabola. This article is currently an experimental vehicle to see how well we can provide proofs and details for a math article without cluttering up the main article itself. See Wikipedia:WikiProject Mathematics/Proofs for some current discussion. This article is "experimental" in the sense that it is a test of one way we may be able to incorporate more detailed proofs in Wikipedia.

Note: Proof is given in a two-column format.


Contents

[edit] 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

[edit] Proof

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

(x − h)2 = 4p(y − k)

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 y = k − p Given
(5) Focus F, which is located at (h,k + p) Given
(6) A parabola with directrix of line l and focus F Given
(7) Point on parabola located at (x,y) 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 P1 on l and one endpoint P2 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 P1 has a y-value of k − p. (4) and (9)
(14) Point P1 has an x-value of x. (7), (9), and (12)
(15) Point P1 is located at (x, k - p). (13) and (14)
(16) Point P2 is located at (x, y). (9)
(17) P_1 P_2 = \sqrt{(x-x)^2 + (y - [k - p])^2} Distance Formula
(18) P_1 P_2 = \sqrt{(y - k + p)^2} Distributive Property
(19) P1P2 = (y − k + p) Apply square root; distance is positive
(20) FP_2 = \sqrt{(x - h)^2 + (y - [k + p])^2} Distance Formula
(21) FP_2 = \sqrt{(x - h)^2 + (y - k - p)^2} Distributive Property
(22) FP2 = P1P2 Definition of Parabola
(23) \sqrt{(x - h)^2 + (y - k - p)^2} = (y - k + p) Substitution
(24) (x − h)2 + (y − k − p)2 = (y − k + p)2 Square both sides
(25) (x − h)2 + k2 + p2 + y2 + 2kp − 2ky − 2py = k2 + p2 + y2 − 2kp − 2ky + 2py Distributive property
(26) (x − h)2 + 2kp − 2py = 2py − 2kp Subtraction Property of Equality
(27) (x − h)2 = 4py − 4kp Addition Property of Equality; Subtraction Property of Equality
(28) (x − h)2 = 4p(y − k) Distributive Property

[edit] Finding the Axis of Symmetry

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 x = h. (31)

[edit] Finding the Vertex

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) (h − h)2 = 4p(y − k) (35); Substitution: (28) and (34)
(37) 0 = 4p(y − k) Simplify
(38) 0 = y − k Division Property of Equality
(39) k = y Addition Property of Equality
(40) y = k Symmetrical Property of Equality
(41) The vertex is located at (h,k). (34) and (40)