InterviewSolution
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InterviewSolution
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Prove that the product of the perpendiculars from `(alpha,beta)`to the pair of lines `a x^2+2h x y+b y^2=0`is`(aalpha^2-2halphabeta+bbeta^2)/(sqrt((a-b)^2+4h^2))` |
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Answer» Let the lines be `y=m_(1)xandy=m_(2)x` or `m_(1)x-y=0andm_(2)x-y=0` were `m_(1)+m_(2)=-(2h)/(b),m_(1)m_(2)=(a)/(b)` Now , if the lengths of perpendicular from (0,0) on these lines are `d_(1)andd_(2)` respectively , then `d_(1)d_(2)=(|m_(1)alpha-beta|)/(sqrt(m_(1)^(2)+1))(|m_(2)alpha-beta|)/(sqrt(m_(2)^(2)+1))` `=(|m_(1)m_(2)alpha^(2)-(m_(1)+m_(2))alpha beta+beta^(2)|)/(sqrt(m_(1)^(2)m_(2)^(2)+m_(1)^(2)+m_(2)^(2)+1))` `=(|m_(1)m_(2)alpha^(2)-(m_(1)+m_(2))alpha beta+beta^(2)|)/(sqrt(m_(1)^(2)m_(2)^(2)+(m_(1)+m_(2))^(2)-2m_(1)m_(2)+1))` `(|(a)/(b)alpha^(2)+(2h)/(b)alpha beta+ beta^(2)|)/(sqrt(a^(2)/(b^(2))+(4h^(2))/(b^(2))-2(a)/(b)+1))` `=(|aalpha+2halpha beta+b beta^(2)|)/(sqrt(a^(2)+4h^(2)-2ab+b^(2)))` `=(|aalpha+2halpha beta+b beta^(2)|)/(sqrt((a-b)^(2)+4h^(2)))` |
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