Ellipses - Conic Sections (II) - Ch.10

II. Conic Sections ELLIPSES (Ch. 10) I. ELLIPSES ↳ an ellipse is the set of all points (x,y) in a plane, the sum of whose distances from two distinct fixed (foci) is constant. (x-h)^2 / a^2 + (y-k)^2 / b^2 = 1 Co vertex d1 d2 focus center focus vertex vertex major aris orientation: horizontal majoraxis center: (h,k) foci (h±c,k) vertices (h±a,k) co-vertices (h,k±b) co-vertex major axis: y=k minor axis: x=h a, b, c relationship: c^2 = a^2 - b^2 / c=√(a^2-b^2) Vertex (x-h)^2 / b^2 + (y-k)^2 / a^2 = 1 focus co-vertex minor axis icenter co-vertex orientation vertical major axis center (h,k) foci (h,k±c) vertices (h,k±a) focus co-vertices: (h±b,k) vertex major axis x=h minor axis: y=x a, b, c relationship c^2 = a^2 - b^2 / a^2-1a^2 c=√(a^2-b^2) II. ECCENTRICITY eccentricity be thought of as the "ovalness" of an ellipse ↳ definition: e=c/a for every ellipse, for ellipses that are nearly circular, e is close to 0 for elongated ellipses, e is close to 1 IV. Conic Sections HYPERBOLAS (Ch.10) I. HYPERBOLAS ↳ a hyperbola is the set of all points (x,y) in a plane for which the absolute value of the difference of the distances from two distinct fired points (foci) is constant   \[\frac{(x-h)^{2}}{a^{2}}-\frac{(y-k)^{2}}{b^{2}}=1\] conjugate axis \[d_{2}\] \[d_{1}\] orientation: horizontal transverse axis transverse Focus vertex center vertex focus transverse axis: y=k conjugate axis: x=h asymptotes: y-k=±(b/a)(x-h) a, b, c relationship c^2=a^2+b^2 \[c=\sqrt{a^{2}+b^{2}}\] transverse axis \[\frac{(y-k)^{2}}{a^{2}}-\frac{(x-h)^{2}}{b^{2}}=1\] foci vertex orientation: vertical transverse axis center (h,k) conjugate axis vertices: (h,k±a) transverse axis: x=h vertex conjugate axis: y=k focus asymptotes: y-k=±(a/b)(x-h) a, b, c relationship c^2=a^2+b^2 \[c=\sqrt{a^{2}+b^{2}}\] asymptote locus asymptote asymptote locus locus asymptote [d_{2}]