Last modified on 24 July 2009, at 15:30

Famous Theorems of Mathematics/Geometry/Conic Sections

Parabola PropertiesEdit

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 P_1 on l and one endpoint P_2 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 P_1 has a y-value of k - p. (4) and (9)
(14) Point P_1 has an x-value of x. (7), (9), and (12)
(15) Point P_1 is located at (x, k - p). (13) and (14)
(16) Point P_2 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) P_1 P_2 = (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) FP_2 = P_1 P_2 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 + k^2 + p^2 + y^2 + 2kp - 2ky - 2py = k^2 + p^2 + y^2 -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

Finding the Axis of SymmetryEdit

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)

Finding the VertexEdit

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)