Geo5.3.8

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Find the centre,eccentricity,length of latusrectum,foci,vertices and equations to the directrices of the hyperbola 9x^{2}-16y^{2}-18x-32y-151=0\,

Structuring the given equation in standard form,we have

9(x-1)^{2}-16(y+1)^{2}-144=0\,

{\frac  {(x-1)^{2}}{16}}-{\frac  {(y+1)^{2}}{9}}=1\,

From the hyperbola equation, we have centre (1,-1)\,

a^{2}=16,b^{2}=9\,

eccentricity 9=16(e^{2}-1),e^{2}={\frac  {25}{16}},e={\frac  {5}{4}}\,

Length of latusrectum {\frac  {2b^{2}}{a}}={\frac  {2\cdot 9}{4}}={\frac  {9}{2}}\,

Foci (h\pm ae,k)=(1\pm 5,-1)=(6,-1),(-4,-1)\,

Vertices (h\pm a,k)=(1\pm 4,-1)=(5,-1),(-3,-1)\,

Equations to directrices x=h\pm {\frac  {a}{e}}=1\pm {\frac  {16}{5}}={\frac  {21}{5}},{\frac  {-11}{5}}\,

5x-21=0,5x+11=0\,

Main Page:Geometry:Hyperbola