Conic section/Proofs

In mathematics, conic sections are relations which represent the equation of the curve (or curves) that result from passing a plane through a cone. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Mathematics, a conic section (or just conic) is a Curve obtained by intersecting a cone (more precisely a circular Conical surface

1 Circles
2 Parabolas
3 Ellipses

Circles

Definition: The locus of all points in a plane which are equidistant from a given point. In Mathematics, a locus ( Latin for "place" plural loci) is a collection of points which share a property This given point is known as the circle's center, and the set distance from the center is known as the radius, represented by the letter r. In Geometry, the centre (or center, in American English of an object is a point in some sense in the middle of the object Remote Authentication Dial In User Service ( RADIUS) is a networking protocol that provides centralized access authorization and accounting management for people or computers

In other words, in a circle with a center (h, k), and a radius of r, a point $(x, y)$ in the circle is r units away from the center. With this, one can insert these variables into the distance formula, which can be modeled by the equation:

$d = \sqrt{(x_1-x_2)^2+(y_1-y_2)^2}$

where 'd' is the distance between two points with coordinates $(x 1, y 1)$ and $(x 2, y 2)$ . Distance is a numerical description of how far apart objects are Because r is the distance between points $(h, k)$ and $(x, y)$ , r can be substituted for r. (x, y) can replace $(x 1, y 1)$ and (h, k) can replace $(x 2, y 2)$ :

$r = \sqrt{(x-h)^2+(y-k)^2}.\,$

By squaring both sides, one is left with the final equation:

$r^2 = (x-h)^2+(y-k)^2.\,$

Parabolas

Definitions

Directrix: line l
Focus: point f which is not contained by line l
Parabola: the locus of points in a plane which are equidistant from line l and a point f
Axis of symmetry: the line which is both perpendicular to the directrix and contains point f
vertex: the locus of points which lie on a the parabola and are points on the axis of symmetry

Proof

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

(x - h) 2 = 4 p (y - 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. In Mathematics, a conic section (or just conic) is a Curve obtained by intersecting a cone (more precisely a circular Conical surface In Geometry, the foci (singular focus) are a pair of special points used in describing Conic sections The four types of conic sections are the Circle In Mathematics, the parabola (pəˈræbələ from the Greek παραβολή) is a Conic section, the intersection of a right circular Reflection symmetry, line symmetry, mirror symmetry, mirror-image symmetry, or bilateral symmetry is Symmetry with respect In the geometry of Curves a vertex is a point of where the first derivative of Curvature is zero	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. Slope is used to describe the steepness incline gradient or grade of a straight 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) $F P 2 = P 1 P 2$	Definition of Parabola
(23) $\sqrt{(x - h)^2 + (y - k - p)^2} = (y - k + p)$	Substitution
(24) $(x - h) 2 + (y - k - p) 2 = (y - k + p) 2$	$S q u a r e b o t h s i d e s$
(25) $(x - h) 2 + k 2 + p 2 + y 2 + 2 k p - 2 k y - 2 p y = k 2 + p 2 + y 2 - 2 k p - 2 k y + 2 p y$	Distributive property
(26) $(x - h) 2 + 2 k p - 2 k y - 2 p y = 2 p y - 2 k p$	Subtraction Property of Equality
(27) $(x - h) 2 = 4 p y - 4 k p$	Addition Property of Equality; Subtraction Property of Equality
(28) $(x - h) 2 = 4 p (y - k)$	Distributive Property

Finding the Axis of Symmetry

Statement	Reason
(29) The axis of symmetry is vertical. Distance is a numerical description of how far apart objects are	(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

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) $(h - h) 2 = 4 p (y - k)$	(35); Substitution: (28) and (34)
(37) $0 = 4 p (y - k)$	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)

Ellipses

For a great discussion of ellipses see the wikipedia article Ellipse

In Mathematics, an ellipse (from the Greek ἔλλειψις literally absence) is a Conic section, the locus of points in a

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