Geo3.1

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Find the equation to the pair of lines passing through the origin and perpendicular to the pair ax^{2}+2hxy+by^{2}=0\, is bx^{2}-2hxy+ay^{2}=0\,

Let the lines be lx_{1}+m_{1}y=0,l_{2}x+m_{2}y=0\,,let these equations be 1 and 2.

Hence (lx_{1}+m_{1}y)(l_{2}x+m_{2}y)\equiv ax^{2}+2hxy+by^{2}\,

Therefore,l_{1}l_{2}=a,m_{1}m_{2}=b,l_{1}m_{2}+l_{2}m_{1}=2h\,

Equations to the lines passing thro' the origin and perpendicular to the 1 and 2 are

m_{1}x-l_{1}y=0,m_{2}x-l_{2}y=0\,

Therefore,the equation to pair of perpendicular lines is (m_{1}x-l_{1}y)(m_{2}x-l_{2}y)=0\,

m_{1}m_{2}x^{2}-xy(l_{1}m_{2}+l_{2}m_{1})+l_{1}l_{2}y^{2}=0\,

bx^{2}-2hxy+ay^{2}=0\,


Main Page:Geometry:Straight Lines-II