# Famous Theorems of Mathematics/Geometry/Conic Sections

## Parabola Properties[edit]

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]

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) |

### 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) | (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) |