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 | edit source]
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)
|
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)
|