Prove that the product of the length of perpendiculars from #(alpha,beta)# to the lines given by #ax^2+2hxy+by^2=0# is #(alpha^2+2halphabeta+b*beta^2)/(sqrt((a-b)^2+4h^2))#?

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Answer 1 · 2018-08-06T07:16:25

Let the equations of the component straight lines are

#y+m_1x=0and y+m_2x=0#

So

#(y+m_1x)(y+m_2x)=y^2+(2h)/bxy+a/bx^2#

#=>y^2+(m_1+m_2)xy+m_2m_2x^2=y^2+(2h)/bxy+a/bx^2#

Comparing we get

#m_1+m_2=(2h)/b#

And

#m_1m_2=a/b#

Now the product of the length of the perpendiculars from the point #(alpha,beta)# is

#=(beta+m_1alpha)/sqrt(1+m_1^2)*(beta+m_2alpha)/sqrt(1+m_2^2)#

#=(beta^2+(m_1+m_2)alphabeta+m_1m_2alpha^2)/sqrt(1+m_1^2+m_2^2+m_1^2m_2^2)#

#=(beta^2+(m_1+m_2)alphabeta+m_1m_2alpha^2)/sqrt(1+(m_1+m_2)^2-2m_1m_2+m_1^2m_2^2)#

#=(beta^2+(2h)/balphabeta+a/balpha^2)/sqrt(1+((2h)/b)^2-2a/b+a^2/b^2)#

#=(aalpha^2+2halphabeta+b*beta^2)/sqrt((a-b)^2+4h^2)#