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Find the equation of the lines which pass through the point of intersection of the pair of lines x^{2}-5xy+4y^{2}+x+2y-2=0\, and are at right angles to them.

From the given a=1,b=4,c=-2,h={\frac  {-5}{2}},g={\frac  {1}{2}},f=1\,

Point of intersection is \left({\frac  {({\frac  {-5}{2}}(1)-4{\frac  {1}{2}})}{1(4)-{\frac  {25}{4}}}},{\frac  {({\frac  {1}{2}}({\frac  {-5}{2}})-1(1)}{4-{\frac  {25}{4}}}}\right)\,


Now the equation to the pair of lines parallel tot he given line passing thro'the origin is x^{2}-5xy+4y^{2}=0\,

Equation to the lines perpendicular to the above line is 4x^{2}+5xy+y^{2}=0\,

Equation to the pair of lines thro'(2,1)\, and perpendicular to the above equation is 4(x-2)^{2}-5(x-2)(y-1)+(y-1)^{2}=0\,

4x^{2}-5xy+y^{2}-21x-12y+27=0\, which is the required perpendicular pair of lines.

Main Page:Geometry:Straight Lines-II