NCERT Solutions
Class 11 Maths
Straight Lines
Ex.10.3 Q.12
Two lines passing through the point (2, 3) intersects each other at an angle of 60°.
If slope of one line is 2, find equation of the other line.
It is given that the slope of the first line, m1 = 2
Let the slope of the other line be m2
The angle between the two lines is 60°.
tan θ = | (m1 – m2) ÷ (1 + m1 × m2) |
=> tan 60° = | (2 - m2) ÷ (1 + 2m2) |
=> √3 = ± (2 - m2) ÷ (1 + 2m2) |
=> √3 = (2 - m2) ÷ (1 + 2m2) or √3 = -(2 - m2) ÷ (1 + 2m2)
=> √3(1 + 2m2) = (2 - m2) or √3(1 + 2m2) = -(2 - m2)
=> √3 + 2√3m2 = 2 - m2 or √3 + 2√3m2 = -2 + m2
=> √3 + 2√3m2 + m2 = 2 or √3 + 2√3m2 - m2 = -2
=> (2√3 + 1) m2 = 2 - √3 or (2√3 – 1) m2 = -2 - √3
=> m2 = (2 - √3) ÷ (2√3 + 1) or m2 = -(2 + √3) ÷ (2√3 - 1)
Case 1: When m2 = (2 - √3) ÷ (2√3 + 1)
The equation of the line passing through point (2, 3) and having a slope of
m2 = (2 - √3) ÷ (2√3 + 1) is given by
(y – 3) = {(2 - √3) ÷ (2√3 + 1)} (x – 2)
=> (y – 3) (2√3 + 1) = (2 - √3) (x – 2)
=> (2√3 + 1) y - 3(2√3 + 1) = (2 - √3) x – 2(2 - √3)
=> (√3 - 2) x + (2√3 + 1) y = -4 + 2√3 + 6√3 + 3
=> (√3 - 2) x + (2√3 + 1) y = -1 + 8√3
In this case, the equation of the other line is (√3 - 2) x + (2√3 + 1) y = -1 + 8√3
Case 2: When m2 = -(2 + √3) ÷ (2√3 - 1)
The equation of the line passing through point (2, 3) and having a slope of
m2 = -(2 + √3) ÷ (2√3 + 1) is given by
(y – 3) = {-(2 + √3) ÷ (2√3 - 1)} (x – 2)
=> (y – 3) (2√3 - 1) = -(2 + √3) (x – 2)
=> (2√3 - 1) y - 3(2√3 - 1) = -(2 + √3) x + 2(2 + √3)
=> (2 + √3) x + (2√3 - 1) y = 4 + 2√3 + 6√3 - 3
=> (2 + √3) x + (2√3 + 1) y = 1 + 8√3
In this case, the equation of the other line is (2 + √3) x + (2√3 + 1) y = 1 + 8√3