An ellipse is the set of all points [latex](x,y)[\/latex] in a plane such that the sum of their distances from two fixed points (called foci) is constant.<\/p>\r\n
Key features of ellipses:<\/strong><\/p>\r\n\r\n Determining orientation:<\/strong><\/p>\r\n\r\n A hyperbola is the set of all points [latex](x,y)[\/latex] in a plane such that the difference of the distances from two fixed points (foci) is a positive constant. A hyperbola has two separate unbounded curves that are mirror images of each other.<\/p>\r\n Key features of hyperbolas:<\/strong><\/p>\r\n\r\n Determining orientation:<\/strong><\/p>\r\n\r\n A parabola<\/strong> is the set of all points [latex](x,y)[\/latex] in a plane that are equidistant from a fixed point (the focus) and a fixed line (the directrix).<\/p>\r\n Key features of parabolas:<\/strong><\/p>\r\n\r\n Determining orientation and direction:<\/strong><\/p>\r\n\r\n The general form of a conic section is [latex]Ax^{2}+Bxy+Cy^{2}+Dx+Ey+F=0[\/latex], where [latex]A[\/latex], [latex]B[\/latex], and [latex]C[\/latex] are not all zero.<\/p>\r\n When [latex]B=0[\/latex] (non-rotated conics):<\/strong><\/p>\r\n\r\n When [latex]B\\neq 0[\/latex] (rotated conics), use the discriminant [latex]B^{2}-4AC[\/latex]:<\/strong><\/p>\r\n The discriminant is invariant (remains unchanged after rotation), enabling identification of the conic type:<\/p>\r\n\r\n Rotation of axes:<\/strong><\/p>\r\n To eliminate the [latex]xy[\/latex] term and rewrite in standard form, rotate the axes by angle [latex]\\theta[\/latex] where [latex]\\cot(2\\theta)=\\frac{A-C}{B}[\/latex]. Use the rotation equations to transform coordinates:<\/p>\r\n\r\n After rotation, the angle [latex]\\theta[\/latex] satisfies:<\/p>\r\n\r\n Degenerate conic sections occur when a plane intersects a double cone through the apex, resulting in a point, a line, or intersecting lines.<\/p>\r\n\r\n Any conic can be defined by a focus, a directrix, and the eccentricity [latex]e[\/latex], which is the ratio [latex]e=\\frac{PF}{PD}[\/latex], where [latex]PF[\/latex] is the distance from a point [latex]P[\/latex] on the conic to the focus and [latex]PD[\/latex] is the distance from [latex]P[\/latex] to the directrix.<\/p>\r\n Eccentricity determines the conic type:<\/strong><\/p>\r\n\r\n Identifying and graphing polar conics:<\/strong><\/p>\r\n\r\n Converting between polar and rectangular form:<\/strong><\/p>\r\n Use the identities [latex]r=\\sqrt{x^{2}+y^{2}}[\/latex], [latex]x=r\\cos\\theta[\/latex], and [latex]y=r\\sin\\theta[\/latex] to convert equations between coordinate systems.<\/p>\r\n\r\n angle of rotation<\/strong><\/p>\r\n An acute angle [latex]\\theta[\/latex] formed by a set of axes rotated from the Cartesian plane where: if [latex]\\cot(2\\theta)>0[\/latex], then [latex]\\theta[\/latex] is between [latex]0\u00b0[\/latex] and [latex]45\u00b0[\/latex]; if [latex]\\cot(2\\theta)<0[\/latex], then [latex]\\theta[\/latex] is between [latex]45\u00b0[\/latex] and [latex]90\u00b0[\/latex]; and if [latex]\\cot(2\\theta)=0[\/latex], then [latex]\\theta=45\u00b0[\/latex].<\/p>\r\n center of an ellipse<\/strong><\/p>\r\n The midpoint of both the major and minor axes of an ellipse.<\/p>\r\n center of a hyperbola<\/strong><\/p>\r\n The midpoint of both the transverse and conjugate axes of a hyperbola, where the axes intersect.<\/p>\r\n central rectangle<\/strong><\/p>\r\n A rectangle centered at the hyperbola's center with sides passing through each vertex and co-vertex; the diagonals of this rectangle are the asymptotes of the hyperbola.<\/p>\r\n conic section (conic)<\/strong><\/p>\r\n Any shape resulting from the intersection of a right circular cone with a plane; includes circles, ellipses, parabolas, and hyperbolas.<\/p>\r\n conjugate axis<\/strong><\/p>\r\n The axis of a hyperbola that is perpendicular to the transverse axis and has the co-vertices as its endpoints.<\/p>\r\n co-vertex<\/strong><\/p>\r\n An endpoint of the minor axis of an ellipse or the conjugate axis of a hyperbola.<\/p>\r\n degenerate conic sections<\/strong><\/p>\r\n Any of the possible shapes formed when a plane intersects a double cone through the apex; types include a point, a line, and intersecting lines.<\/p>\r\n directrix<\/strong><\/p>\r\n A fixed line used in the definition of a conic section; for a parabola, all points on the curve are equidistant from the focus and the directrix. More generally, a line such that the ratio of the distance from points on the conic to the focus to the distance to the directrix is constant (the eccentricity).<\/p>\r\n\r\n \t
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Hyperbolas<\/h3>\r\n
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Parabolas<\/h3>\r\n
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Rotation of Axes<\/h3>\r\n
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Conic Sections in Polar Coordinates<\/h3>\r\n
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Key Equations<\/h2>\r\n
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\r\n Horizontal ellipse, center at (h,k)<\/strong><\/td>\r\n [latex]\\frac{(x-h)^{2}}{a^{2}}+\\frac{(y-k)^{2}}{b^{2}}=1[\/latex], where [latex]a>b[\/latex]<\/td>\r\n<\/tr>\r\n \r\n Vertical ellipse, center at (h,k)<\/strong><\/td>\r\n [latex]\\frac{(x-h)^{2}}{b^{2}}+\\frac{(y-k)^{2}}{a^{2}}=1[\/latex], where [latex]a>b[\/latex]<\/td>\r\n<\/tr>\r\n \r\n Relationship for ellipse foci<\/strong><\/td>\r\n [latex]c^{2}=a^{2}-b^{2}[\/latex]<\/td>\r\n<\/tr>\r\n \r\n Hyperbola, center at (h,k), transverse axis parallel to x-axis<\/strong><\/td>\r\n [latex]\\frac{(x-h)^{2}}{a^{2}}-\\frac{(y-k)^{2}}{b^{2}}=1[\/latex]<\/td>\r\n<\/tr>\r\n \r\n Hyperbola, center at (h,k), transverse axis parallel to y-axis<\/strong><\/td>\r\n [latex]\\frac{(y-k)^{2}}{a^{2}}-\\frac{(x-h)^{2}}{b^{2}}=1[\/latex]<\/td>\r\n<\/tr>\r\n \r\n Relationship for hyperbola foci<\/strong><\/td>\r\n [latex]c^{2}=a^{2}+b^{2}[\/latex]<\/td>\r\n<\/tr>\r\n \r\n Asymptotes for hyperbola (horizontal transverse axis)<\/strong><\/td>\r\n [latex]y=\\pm\\frac{b}{a}(x-h)+k[\/latex]<\/td>\r\n<\/tr>\r\n \r\n Asymptotes for hyperbola (vertical transverse axis)<\/strong><\/td>\r\n [latex]y=\\pm\\frac{a}{b}(x-h)+k[\/latex]<\/td>\r\n<\/tr>\r\n \r\n Parabola, vertex at (h,k), axis of symmetry parallel to x-axis<\/strong><\/td>\r\n [latex](y-k)^{2}=4p(x-h)[\/latex]<\/td>\r\n<\/tr>\r\n \r\n Parabola, vertex at (h,k), axis of symmetry parallel to y-axis<\/strong><\/td>\r\n [latex](x-h)^{2}=4p(y-k)[\/latex]<\/td>\r\n<\/tr>\r\n \r\n General form of a conic<\/strong><\/td>\r\n [latex]Ax^{2}+Bxy+Cy^{2}+Dx+Ey+F=0[\/latex]<\/td>\r\n<\/tr>\r\n \r\n Discriminant for identifying conics<\/strong><\/td>\r\n [latex]B^{2}-4AC[\/latex]<\/td>\r\n<\/tr>\r\n \r\n Angle of rotation<\/strong><\/td>\r\n [latex]\\cot(2\\theta)=\\frac{A-C}{B}[\/latex]<\/td>\r\n<\/tr>\r\n \r\n Rotation of a conic section<\/strong><\/td>\r\n [latex]x=x'\\cos\\theta-y'\\sin\\theta[\/latex] and [latex]y=x'\\sin\\theta+y'\\cos\\theta[\/latex]<\/td>\r\n<\/tr>\r\n \r\n Polar equation (directrix x = \u00b1p)<\/strong><\/td>\r\n [latex]r=\\frac{ep}{1\\pm e\\cos\\theta}[\/latex]<\/td>\r\n<\/tr>\r\n \r\n Polar equation (directrix y = \u00b1p)<\/strong><\/td>\r\n [latex]r=\\frac{ep}{1\\pm e\\sin\\theta}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n Glossary<\/h2>\r\n