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`CONIC SECTIONS:YPARABOLALine of symmetry: x=h 2p (h, k+p) p 2pFocus(h, k) p. .VertexDirectrixy=k-p XStandard Equation Vertical Horizontal (x ­ h)2 = 4p(y ­ k) (y ­ k)2 = 4p(x ­ h)Vertex (h, k) (h, k)Focus (h, k + p) (h + p, k)Directrix y=k­p x=h­pEvery point on a parabola is the same distance from the focus as from the directrix.Y d1 2p p.Pd2 2p (h+p, k)d1 = d2 Line of symmetry: y=k(h, k). . .pFocus VertexDirectrixx=h-p XDepartment of Mathematics, Sinclair Community College, Dayton, OH1CONIC SECTIONS:P d1ELLIPSE.c(h, k+b)d2 Center (h, k)Vertex(h-a, k)Vertex(h+c, k) a.(h-c, k)....(h+a, k)FocusMinor Axis(Length = 2b)bMajor Axis(Length = 2a)Focus(h, k-b)Standard Equation Horizontal Vertical Eccentricity: Center (h, k) (h, k) Foci (h+c, k), (h-c, k) (h, k+c), (h, k-c) c2 = a2 ­ b2(graphs not to scale)Vertices (h+a, k), (h-a, k) (h, k+a), (h, k-a)(x - h) (y - k ) + =1 a2 b2 (x - h) 2 (y - k ) 2 + =1 b2 a22 2c e= &lt;1 aa&gt;bFor any point P on an ellipse, the sum of the distances from P to the foci is a constant: d1 + d2 = 2a. If a = b, then c = 0, and the ellipse is a circle.(h, k+a)Focus. . . . .Vertex(h, k+c)c Center (h, k) (h-b, k) b a (h+b, k)(h, k-c)FocusVertex2Department of Mathematics, Sinclair Community College, Dayton, OH(h, k-a)CONIC SECTIONS:HYPERBOLAConjugate Axis x=hP d1Focus·Vertexd2a(h-c, k).(h-a, k).c(h, k) Center. .bVertex(h+a, k)(h+c, k).FocusTransverse Axis y=ky = k + (b/a)(x ­ h)y = k ­ (b/a)(x ­ h)Standard Equation Transverse axis parallel to x-axis Transverse axis parallel to y-axis Eccentricity:Center (h, k)Foci (h+c, k), (h­c, k) (h, k+c), (h, k-c) c2 = a2 + b2(graphs not to scale)Vertices (h+a, k), (h-a, k)Slant Asymptotes y=k±  y=k± (x - h) (y - k ) - =1 2 a b22 2b  (x-h) a a  (x-h) b(y - k ) 2 (x - h) 2 - =1 a2 b2(h, k)(h, k+a), (h, k-a)c e= &gt;1 aFor any point P on a hyperbola, the difference of the distances of P to the foci is a constant: d2 ­ d1 = 2aTransverse Axis x=h(h, k+c)y = k ­ (a/b)(x-h)c(h, k+a)(h, k)y = k + (a/b)(x-h)(h, k-a) (h, k-c) Focus. . . . .FocusVertexaConjugate Axis y=kbVertexDepartment of Mathematics, Sinclair Community College, Dayton, OH3Another important type of hyperbola is defined by the equation xy = C, where C is a constant. The asymptotes are the x and y axes. The function is symmetric about the origin, and the transverse axis is y = x. (If C is negative, the transverse axis is y = -x, and the graph is in Quadrants II and IV.) This graph can also be viewed as a standard, x-y oriented hyperbola that has been rotated 45° into a new coordinate system.. Focus: ( 2C , 2C ) . Vertex: ( C , C ) . .____________________________________________________________________________THE SECOND-DEGREE EQUATIONThe ellipse/circle, parabola, and hyperbola functions are all cases of the second-degree equation: Ax2 + Bxy + Cy2 + Dx + Ey + F = 0. Ax2 A Circle Ellipse Parabola Hyperbola Center/vertex not at origin Rotated axis  0 =C  C = 0 if C  0 opp. sign of C + Bxy + B =0 =0 =0 =0 Cy2 C =A  A = 0 if A  0 opp. sign of A  0 and/or  0 + Dx D + Ey E + F FTo rotate axes (eliminate the xy term):  = angle of rotation from x-axis cot 2 = (A ­ C)/B x = x'cos ­ y'sin y = x'sin + y'cos Substitute  A'(x')2 + C'(y')2 + D'x' + E'y' + F' = 0 Invariants: F' = F A + C = A' + C' B2 ­ 4AC = (B')2 ­ 4A'C' B2 ­ 4AC &lt; 0  ellipse/circle B2 ­ 4AC = 0  parabola B2 ­ 4AC &gt; 0  hyperbolaDepartment of Mathematics, Sinclair Community College, Dayton, OH4`

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