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: and that the vertex of this parabola is (h,k) StatementReason(1) Arbitrary real value hGiven(2) Arbitrary real value kGiven(3) Arbitrary real value p where p is not equal to 0Given(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 FGiven(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 StatementReason(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) Finding the Vertex StatementReason(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)