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

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

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

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

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

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

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