Geo3.27

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If the chord x+y=b\, of the curve x^{2}+y^{2}-2ax-4a^{2}=0\, subtends a right angle at the origin ,prove that b(b-a)=4a^{2}\,

Equation of the chord is x+y=b,{\frac  {x+y}{b}}=1\,

Homogenising the equation of the curve with equation 1

x^{2}+y^{2}-2ax({\frac  {x+y}{b}}-4a^{2}[{\frac  {x+y}{b}}]^{2}=0\,

In this pair of lines are at right angles,then

[1-{\frac  {2a}{b}}-{\frac  {4a^{2}}{b^{2}}}]+[1-{\frac  {4a^{2}}{b^{2}}}]=0\,

2b^{2}-2ab-8a^{2}=0,b(b-a)=4a^{2}\,


Main Page:Geometry:Straight Lines-II