Parabola Properties [ edit ]
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 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 $x=h$ .
(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) $(h-h)^{2}=4p(y-k)$
(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)