19. Find the equations to the strai...
Question

# 19. Find the equations to the straight lines which pass through origin and are inclined at 75° to the straight line x + y + 3(y - x) = a

JEE/Engineering Exams
Maths
Solution
87
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( frac{1+sqrt{3}}{1-sqrt{3}}=frac{m_{1}-m_{2}}{1+m_{1} m} ) Prurs ( Rightarrow quad 1 quad=quad m_{1}-m ) ( 2 n ) 1) for a line having slope ( m_{1} ) and othere lein having Slope ( m_{2} ), the acute ongle between ( Rightarrow quad m_{1} m_{2}+1=m_{1}^{2}-m_{1} m_{2} ) (2) for a luin having ( 2 M y ) ( =-frac{9}{b} ) (3) Point slobe equater of buic in ( m_{L}=1-sqrt{3} quad-1(1+sqrt{3}) ) Slope equatuin of luil in ( y-y_{1}=mleft(x-x_{1}right) quad ) i foint ( left(x_{1}+x_{1}right) ) ( m_{2}=1+3-2 sqrt{3}-(1+3+2 sqrt{3} ) (1) Sbe of luie ( x+y+sqrt{3}(y-x)-a=0 quad 2(1+sqrt{3})(1-sqrt{3}) ) ( =(sqrt{3}+1) y+(sqrt{3}-1) x-a=0 ) ( Rightarrow(sqrt{3}-1) x+(sqrt{3}+1) y-a=0 ) ( m_{1}=frac{-(sqrt{3}-1)}{sqrt{3}+1}=frac{1-sqrt{3}}{1+sqrt{3}} ) (2) Whe slope of Required luic is ( m_{2} ) ( m_{2}=-4 sqrt{3} ) begin{tabular}{l|l|lllll} hline ( operatorname{an}(80+45) ) & ( = ) & ( m_{1}-m_{2} ) & & & hline & ( 1+m_{1} m_{2} ) & & & & & hline end{tabular} ( frac{sqrt{3}+1}{frac{1}{sqrt{3}}-1}=frac{m_{1}-m_{2}}{1+m_{1} m_{2}} ) ( -0=sqrt{3}(x-0) ) ( =sqrt{3} x )