Two lines passing through the point (2, 3) intersect each other at an angle of 60 0 . If the slope of one line is 2. Find the equation of the other line.
Answers
Answered by
27
let m1 and m2 are the slope of two lines
we know ,
tan@=mod {(m1-m2)/(1+m1.m2)}
where @ is angle between two lines
hence,
tan60=mod {(2-m2)/(1+2m2)}
root3mod {(1+2m2)}=mod {2-m2}
root3 (1+2m2)=+_(2-m2)
root3+2root3 m2=+_(2-m2)
hence
m2=(2-root3)/(1+2root3) , (2+root3)/(1-2root3)
now equation of line ,
(y-3)=m2 (x-2)
put m2 value and equation
(1+2root3)(y-3)=(2-root3)(x-2)
or,
(1-2root3)(y-3)=(2+root3)(x-2)
we know ,
tan@=mod {(m1-m2)/(1+m1.m2)}
where @ is angle between two lines
hence,
tan60=mod {(2-m2)/(1+2m2)}
root3mod {(1+2m2)}=mod {2-m2}
root3 (1+2m2)=+_(2-m2)
root3+2root3 m2=+_(2-m2)
hence
m2=(2-root3)/(1+2root3) , (2+root3)/(1-2root3)
now equation of line ,
(y-3)=m2 (x-2)
put m2 value and equation
(1+2root3)(y-3)=(2-root3)(x-2)
or,
(1-2root3)(y-3)=(2+root3)(x-2)
Answered by
17
Two lines are possible.
P(2,3)
Given L1 : slope = 2
Let L2 be : y = m x + c
case 1)
Tan 60° = (m1 - m2) / [1 + m1 m2]
√3 = (2 - m) / (1 + 2 m)
2 - m = √3 + 2√3 m
m = (2 - √3) / (2√3 + 1) = (2-√3)(2√3 -1) / 11
m = (5√3 - 8) / 11
So : L2 : y = (5√3 - 8) /11 * x + c
P(2,3) lies on it.
3 = (5√3 - 8)/11 * 2 + c
c = (49 - 10√3) / 11
So ans: 11 y = (5 √3 - 8) x + (49 - 10√3)
case 2: angle = -60 deg
Tan (-60°) = (m1 - m2) / [1 + m1 m2]
-√3 = (2 - m) / (1 + 2 m)
2 - m = -√3 - 2√3 m
m = - (2 + √3) / (2√3 - 1) = - (2+√3)(2√3+1) / 11
m = - (5√3 + 8) / 11
So : L2 : y = - (5√3 + 8) /11 * x + c
P(2,3) lies on it.
3 = - (5√3 + 8)/11 * 2 + c
c = (49 + 10√3) / 11
So ans: 11 y = - (5 √3 + 8) x + (49 + 10√3)
P(2,3)
Given L1 : slope = 2
Let L2 be : y = m x + c
case 1)
Tan 60° = (m1 - m2) / [1 + m1 m2]
√3 = (2 - m) / (1 + 2 m)
2 - m = √3 + 2√3 m
m = (2 - √3) / (2√3 + 1) = (2-√3)(2√3 -1) / 11
m = (5√3 - 8) / 11
So : L2 : y = (5√3 - 8) /11 * x + c
P(2,3) lies on it.
3 = (5√3 - 8)/11 * 2 + c
c = (49 - 10√3) / 11
So ans: 11 y = (5 √3 - 8) x + (49 - 10√3)
case 2: angle = -60 deg
Tan (-60°) = (m1 - m2) / [1 + m1 m2]
-√3 = (2 - m) / (1 + 2 m)
2 - m = -√3 - 2√3 m
m = - (2 + √3) / (2√3 - 1) = - (2+√3)(2√3+1) / 11
m = - (5√3 + 8) / 11
So : L2 : y = - (5√3 + 8) /11 * x + c
P(2,3) lies on it.
3 = - (5√3 + 8)/11 * 2 + c
c = (49 + 10√3) / 11
So ans: 11 y = - (5 √3 + 8) x + (49 + 10√3)
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