Angle Between Two lines | Mathematics (Maths) Class 11 - Commerce

E. Angle Between Two Straight Lines

Ifθ is the acute angle between two lines, then   where  m₁ and m₂ are the slopes of the two lines and are finite.

Notes :

(i) If the two lines are perpendicular to each other then m₁m₂ = -1

(ii) Any line perpendicular to ax + by + c = 0 is of the form bx - ay + k = 0

(iii) If the two lines are parallel or coincident, then m₁= m₂

(iv)  Any line parallel to ax + by + c = 0 is of the form ax + by + k = 0

(v) If any of the two lines is perpendicular to x-axis, then the slope of that line is infinite.

 Let m₁ = ∝ ,    or θ = |90° – a |, where tan a = m₂

i.e. angle θ is the complimentary to the angle which the oblique line makes with the x-axis.

(vi) If lines are equally inclined to the coordinate axis then m₁ + m₂ = 0

Ex.11 Find the equation to the straight line which is perpendicular bisector of the line segment AB, where A, B are (a,b) and (a', b') respectively.

Sol. Equation of AB is y - b =

i.e. y (a' - a) - x (b' - b) = a'b - ab'.

Equation to the line perpendicular to AB is of the form (b' - b)y + (a' - a)x + k = 0        ....(1)

Since the midpoint of AB lies on (1),

Hence the required equation of the straight line is

(1) Equation of straight Lines passing through a given point and equally inclined to a given line : 

Let the straight passing through the point (x₁, y₁) and make equal angles with the given straight line y = mx + c. If m is the slope of the required line and a is the angle which this line makes with the given line then

(2) The above expression for tana, given two values of m, say m_A and m_B.

The required equations of the lines through the point (x₁, y₁) and making equal angles a with the given line are

y - y₁ = m_A(x - x₁), y - y₁ = m_B (x - x₁)

Ex.12 Find the equation to the sides of an isosceles right-angled triangled, the equation of whose hypotenuse is 3x + 4y = 4 and the opposite vertex is the point (2, 2).

Sol. The problem can be restarted as :

Find the equation to the straight lines passing through the given point (2, 2) and making equal angles of 45° with the given straight line 3x + 4y - 4 = 0. Slope of the line 3x + 4y - 4 = 0 is

⇒ ,

i.e.,

m_A = , and m_B = - 7

Hence the required equations of the two lines are

y - 2 = m_A(x - 2) and y - 2 = m_B (x - 2) ⇒ 7y - x - 12 = 0 and 7x + y = 16.