Geo5.3.7

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Find the centre,eccentricity,length of latusrectum,foci,vertices and equations to the directrices of the hyperbola {\frac  {(y-4)^{2}}{9}}-{\frac  {(x-5)^{2}}{16}}=1\,

For the given hyperbola,b^{2}=9,a^{2}=16,(h,k)=(5,4)\,

Centre is (5,4)\,

eccentricity {\frac  {{\sqrt  {a^{2}+b^{2}}}}{b}}={\frac  {{\sqrt  {9+16}}}{3}}={\frac  {5}{3}}\,

Length of latusrectum is {\frac  {2a^{2}}{b}}={\frac  {2(16)}{3}}={\frac  {32}{3}}\,

Foci (h,k\pm be)=(5,4\pm 5)=(5,9),(5,-1)\,

Vertices (h,k\pm b)=(5,4\pm 3)=(5,7),(5,1)\,

Equations to directrices y=k\pm {\frac  {b}{e}}=4\pm {\frac  {9}{5}}={\frac  {29}{5}},{\frac  {11}{5}}\,

5y-29=0,5y-11=0\,


Main Page:Geometry:Hyperbola