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

E. Angle Between Two Straight Lines

Ifθ is the acute angle between two lines, then  Angle Between Two lines | Mathematics (Maths) Class 11 - Commerce where  mand m2 are the slopes of the two lines and are finite.

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

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

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

Hence the required equation of the straight line is Angle Between Two lines | Mathematics (Maths) Class 11 - Commerce

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

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

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

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

y - y1 = mA(x - x1), y - y1 = mB (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 :

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

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

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

i.e., Angle Between Two lines | Mathematics (Maths) Class 11 - Commerce

mA = Angle Between Two lines | Mathematics (Maths) Class 11 - Commerce, and mB = - 7

Hence the required equations of the two lines are

y - 2 = mA(x - 2) and y - 2 = mB (x - 2) ⇒ 7y - x - 12 = 0 and 7x + y = 16.

The document Angle Between Two lines | Mathematics (Maths) Class 11 - Commerce is a part of the Commerce Course Mathematics (Maths) Class 11.
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FAQs on Angle Between Two lines - Mathematics (Maths) Class 11 - Commerce

1. What is the formula to find the angle between two lines?
Ans. The formula to find the angle between two lines is given by the equation: $$\theta = \arctan\left(\frac{m_2 - m_1}{1 + m_1 \cdot m_2}\right)$$ where $m_1$ and $m_2$ are the slopes of the two lines.
2. How do I find the slope of a line given its equation?
Ans. To find the slope of a line given its equation, you can use the following steps: 1. Write the equation of the line in slope-intercept form, i.e., $y = mx + b$, where $m$ is the slope. 2. The coefficient of $x$ in the equation represents the slope of the line.
3. Can the angle between two lines be negative?
Ans. No, the angle between two lines cannot be negative. The angle is always measured as a positive value between 0 and 180 degrees. If the angle between two lines is negative, it means that the lines are pointing in opposite directions.
4. How can I calculate the angle between two lines in degrees?
Ans. To calculate the angle between two lines in degrees, you can use the formula: $$\theta = \arctan\left(\frac{m_2 - m_1}{1 + m_1 \cdot m_2}\right)$$ Once you have the angle in radians, you can convert it to degrees by multiplying it by $\frac{180}{\pi}$.
5. What does it mean if the angle between two lines is 90 degrees?
Ans. If the angle between two lines is 90 degrees, it means that the lines are perpendicular to each other. Perpendicular lines have slopes that are negative reciprocals of each other. This means that the product of their slopes is -1.
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