Conic Sections Circle Definition A circle is the

Conic Sections

Circle Definition: A circle is the set of all points in a plane at a fixed positive distance (radius)from a fixed point (center).

Case(1) : Equation Center Radius x 2 + y 2 = r 2 (0, 0) r

Case(2) : Equation Center Radius (x-h)2 + (y-k)2= r 2 (h, k) r

Parabola Definition: A parabola is the set of all points in a plane equidistant from a fixed point (focus) and a fixed line (directrix) not containing the focus.

Case(1) : Equation Axis y 2 = 4 cx X-axis Vertex Focus Directrix |LR| End points of LR (0, 0) (c, 0) x=-c 4|c| (c, 2 c) ^ (c, -2 c)

Case(2) : Equation Axis x 2 = 4 cy y-axis Vertex Focus Directrix (0, 0) (0, c) y=-C |LR| End points of LR 4|c| (2 c, c) ^ (-2 c, c)

c) Horizontal and Vertical Shifting: Equation 1 (x-h)2 = 4 c(y-k) Axis // Y-axis Vertex Focus Directrix |LR| (h, k) (h, k+c) Y=k-c 4|c| End points of LR (h+2 c, k+c) ^ (h-2 c, k+c)

Equation 2 (y-k)2 = 4 c(x-h) Axis // X-axis Vertex Focus Directrix |LR| (h, k) (h+c, k) X=h-C 4|c| End points of LR (h+c, k+2 c) ^ (h+c, k-2 c)

Ellipse Definition: An ellipse is the set of all points (x, y) such that the sum of the distances from (x, y) to a pair of distinct fixed points (foci) is a fixed constant.

Case(1) : eccentricity ( Where (b 2 = a 2 – c 2) ) Major Axis Minor axis Center Vertices Foci Covertices x-axis y-axis (0, 0) (±a, 0) (±c, 0) (0, ±b) |L. R| End points of L. R (±c, ± )

Case(2) : eccentricity ( Where (b 2 = a 2 – c 2) ) Major Axis Minor axis Center Vertices Foci Covertices y-axis x-axis (0, 0) (0, ±a) (0, ±c) (±b, 0) |L. R| End points of L. R (± , ±c)

Case (3)Shifting: Horizontal and Vertical Shifting I : eccentricity ( ) Where (b 2 = a 2 – c 2) Major Axis Minor Vertice Center axis s // x-axis // y-axis (h, k) Foci (h±a, k) (h±c, k) Covertice s (h, k±b) |L. R| End points of L. R (h±c , k± )

Horizontal and Vertical Shifting II : eccentricity ( ) Where (b 2 = a 2 – c 2) Major Axis Minor Vertice Center axis s // y-axis // x-axis (h, k) Foci (h, k±a) (h, k±c) Covertice s (h±b, k) |L. R| End points of L. R (h± , k±c)

Hyperbola Definition: A hyperbola is the set of all points (x, y) in a plane such that positive difference between the distances from (x, y) to a pair of distinct fixed points (foci) is a fixed constant.

Case(1) : Transitive axis Conjugate axis center vertices foci x-axis y-axis (0, 0) (a, 0)^(-a, 0) (c, 0)^(-c, 0) asymptote s End point of LR eccentricity Covertices (0, b)^(0, -b)

Case(2) : Transitive axis Conjugate axis center vertices foci y-axis x-axis (0, 0) (0, a) ^ (0, -a) (0, c) ^ (0, -c) asymptote s End point of LR eccentricity Covertices (b, 0) ^ (-b, 0)

Case (3)Shifting: Horizontal and Vertical Shifting I : Transitive axis Conjugate axis center vertices foci // x-axis // y-axis (h, k) (h-a, k)^(h+a, k) (h-c, k)^(h+c, k) Covertices (h, k-b)^(h, k+b) asymptotes End point of LR eccentricity

Horizontal and Vertical Shifting II : Transitive axis Conjugate axis center vertices foci // y-axis // x-axis (h, k) (h, k-a)^ (h, k+a) (h, k+c) ^(h, k-c) asymptotes End point of LR eccentricity Covertices (h+b, k)^(h-b, k)