# Famous Theorems of Mathematics/Geometry/Conic Sections

## Parabola Properties

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

${\displaystyle (x-h)^{2}=4p(y-k)}$

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 ${\displaystyle y=k-p}$ Given
(5) Focus F, which is located at ${\displaystyle (h,k+p)}$ Given
(6) A parabola with directrix of line l and focus F Given
(7) Point on parabola located at ${\displaystyle (x,y)}$ 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 ${\displaystyle P_{1}}$ on l and one endpoint ${\displaystyle P_{2}}$ 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 ${\displaystyle P_{1}}$ has a y-value of ${\displaystyle k-p}$. (4) and (9)
(14) Point ${\displaystyle P_{1}}$ has an x-value of x. (7), (9), and (12)
(15) Point ${\displaystyle P_{1}}$ is located at (x, k - p). (13) and (14)
(16) Point ${\displaystyle P_{2}}$ is located at (x, y). (9)
(17) ${\displaystyle P_{1}P_{2}={\sqrt {(x-x)^{2}+(y-[k-p])^{2}}}}$ Distance Formula
(18) ${\displaystyle P_{1}P_{2}={\sqrt {(y-k+p)^{2}}}}$ Distributive Property
(19) ${\displaystyle P_{1}P_{2}=(y-k+p)}$ Apply square root; distance is positive
(20) ${\displaystyle FP_{2}={\sqrt {(x-h)^{2}+(y-[k+p])^{2}}}}$ Distance Formula
(21) ${\displaystyle FP_{2}={\sqrt {(x-h)^{2}+(y-k-p)^{2}}}}$ Distributive Property
(22) ${\displaystyle FP_{2}=P_{1}P_{2}}$ Definition of Parabola
(23) ${\displaystyle {\sqrt {(x-h)^{2}+(y-k-p)^{2}}}=(y-k+p)}$ Substitution
(24) ${\displaystyle (x-h)^{2}+(y-k-p)^{2}=(y-k+p)^{2}}$ Square both sides
(25) ${\displaystyle (x-h)^{2}+k^{2}+p^{2}+y^{2}+2kp-2ky-2py=k^{2}+p^{2}+y^{2}-2kp-2ky+2py}$ Distributive property
(26) ${\displaystyle (x-h)^{2}+2kp-2py=2py-2kp}$ Subtraction Property of Equality
(27) ${\displaystyle (x-h)^{2}=4py-4kp}$ Addition Property of Equality; Subtraction Property of Equality
(28) ${\displaystyle (x-h)^{2}=4p(y-k)}$ Distributive Property

### 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 ${\displaystyle x=h}$. (31)

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