A straight line passes through the point A(1,2) and makes an angle theta with the positive direction of x axes. find theta,given that the distance of A from the point of intersection of this line with the line x+y=4 is root5/3 unit?

1 Answer
P dilip_k
Aug 11, 2017

given that a straight line passes through the point A(1,2) and makes an angle #theta# with the positive direction of x axes, we can write its equation as

#y-2=(x-1)tantheta ......[1]#

Solving this equation with equation

#x+y=4......[2]#

Subtracting [1] from[2]

#x+2=4-(x-1)tantheta#

#=>x+2=4-xtantheta+tantheta#

#=>x+xtantheta=2+tantheta#

#=>x=(2+tantheta)/(1+tantheta)#

So #y=4-x=4-(2+tantheta)/(1+tantheta)=(2+3tantheta)/(1+tantheta)#

So point of intersection of [1] and [2] is

#((2+tantheta)/(1+tantheta),(2+3tantheta)/(1+tantheta))#

Given that the distance of A from the point of intersection of this line with the line x+y=4 is #sqrt5/3# unit, we can write

#((2+tantheta)/(1+tantheta)-1)^2+((2+3tantheta)/(1+tantheta)-2)^2=(sqrt5/3)^2#

#=>1/(1+tantheta)^2+tan^2theta/(1+tantheta)^2=5/9#

#=>(1+tan^2theta)/(1+tantheta)^2=5/9#

#=>sec^2theta/(1+tantheta)^2=5/9#

#=>sectheta/(1+tantheta)=sqrt5/3#

#=>1/(costheta+sintheta)=sqrt5/3#

#=>costheta+sintheta=3/sqrt5#

#=>1/sqrt2costheta+1/sqrt2sintheta=3/sqrt10#

#=>cos(theta-45^@)=3/sqrt10=cos18.4^@#

#=>theta= 63.4^@#

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