Geo5.3.34

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The asymptotes of the hyperbola are parallel to the lines 2x+3y=0,3x+2y=0\,. Its centre is at (1,2)\, and passes thro' the point (5,3)\,.Find its equation.

The equations of the asymptotes are 2x+3y+k=0,3x+2y+l=0\,

The asymptotes pass through the centre (1,2)\,

2(1)+3(2)+k=0,3(1)+2(2)+l=0,k=-8,l=-7\,

Therefore,the asymptotes are 2x+3y-8=0,3x+2y-7=0\,

The combined equation of the asymptotes is (2x+3y-8)(3x+2y-7)=0\,

Hence the equation to the hyperbola is (2x+3y-8)(3x+2y-7)+\lambda =0\,

Since the hyperbola passes through (5,3)\,

(10+9-8)(15+6-7)+\lambda =0,\lambda =-154\,

Hence the equation of hyperbola is (2x+3y-8)(3x+2y-7)-154=0\,

6x^{2}+13xy+6y^{2}-38x-37y-98=0\,


Main Page:Geometry:Hyperbola