conic section

conic section

conic section or conic, curve formed by the intersection of a plane and a right circular cone (conical surface). The ordinary conic sections are the circle, the ellipse, the parabola, and the hyperbola. When the plane passes through the vertex of the cone, the result is a point, a straight line, or a pair of intersecting straight lines; these are called degenerate conic sections. There are many examples of the conic sections, both in nature and in technology. The orbits of planets and satellites are elliptical, and parallel reflectors (e.g., in telescopes) are parabolic in shape.

Any two-dimensional curve traced by the intersection of a right circular cone with a plane. If the plane is perpendicular to the cone's axis, the resulting curve is a circle. Intersections at other angles result in ellipses, parabolas, and hyperbolas. The conic sections are studied in Euclidean geometry to analyze their physical properties and in analytic geometry to derive their equations. In either context, they have useful applications to optics, antenna design, structural engineering, and architecture.

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In mathematics, conic sections are relations which represent the equation of the curve (or curves) that result from passing a plane through a cone.

Circles

Definition: The locus of all points in a plane which are equidistant from a given point. 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 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). 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 (xy) on a parabola with vertex (h,k), focus (hk + p), and directrix y = k − p:

(x - h)^2 = 4p(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. 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 - 2ky - 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

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

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)

Ellipses

For a great discussion of ellipses see the wikipedia article Ellipse

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