RD Chapter 23 The Straight Lines Ex 23.1 Solutions
Question - 1 : - Find the slopes of the lines which make the following angles with the positive direction of x – axis:
(i) – π/4
(ii) 2π/3
Answer - 1 : -
(i) – π/4
Let the slope of the line be ‘m’
Where, m = tan θ
So, the slope of Line is m = tan (– π/4)
= – 1
∴ The slope of the line is – 1.
(ii) 2π/3
Let the slope of the line be ‘m’
Where, m = tan θ
So, the slope of Line is m = tan (2π/3)
∴ The slope of the line is –√3
Question - 2 : - Find the slopes of a line passing through the following points :
(i) (–3, 2) and (1, 4)
(ii) (at21, 2at1) and (at22,2at2)
Answer - 2 : -
(i) (–3, 2) and (1, 4)
By using the formula,
∴ The slope of the lineis ½.
(ii) (at21,2at1) and (at22, 2at2)
By using the formula,
Question - 3 : - State whether the two lines in each of the following are parallel, perpendicular or neither:
(i) Through (5, 6) and (2, 3); through (9, –2) and (6, –5)
(ii) Through (9, 5) and (– 1, 1); through (3, –5) and (8, –3)
Answer - 3 : -
(i) Through (5, 6) and (2,3); through (9, –2) and (6, –5)
By using the formula,
So, m2 =1
Here, m1 = m2 =1
∴ The lines areparallel to each other.
(ii) Through (9, 5) and (–1, 1); through (3, –5) and (8, –3)
By using the formula,
Question - 4 : - Find the slopes of a line
(i) which bisects the first quadrant angle
(ii) which makes an angle of 300 with the positive direction of y – axis measured anticlockwise.
Answer - 4 : -
(i) Which bisects thefirst quadrant angle?
Given: Line bisectsthe first quadrant
We know that, if theline bisects in the first quadrant, then the angle must be between line and thepositive direction of x – axis.
Since, angle = 90/2 =45o
By using the formula,
The slope of the line,m = tan θ
The slope of the linefor a given angle is m = tan 45o
So, m = 1
∴ The slope of the lineis 1.
(ii) Which makes anangle of 300 with the positive direction of y – axis measuredanticlockwise?
Given: The line makesan angle of 30o with the positive direction of y – axis.
We knowthat, angle between line and positive side of axis => 90o +30o = 120o
By using the formula,
The slope of the line,m = tan θ
The slope of the linefor a given angle is m = tan 120o
So, m = – √3
∴ The slope of the lineis – √3.
Question - 5 : - Using the method of slopes show that the following points are collinear:
(i) A (4, 8), B (5, 12), C (9, 28)
(ii) A(16, – 18), B(3, – 6), C(– 10, 6)
Answer - 5 : -
(i) A (4, 8), B (5, 12), C(9, 28)
By using the formula,
The slope of the line= [y2 – y1] / [x2 – x1]
So,
The slope of line AB =[12 – 8] / [5 – 4]
= 4 / 1
The slope of line BC =[28 – 12] / [9 – 5]
= 16 / 4
= 4
The slope of line CA =[8 – 28] / [4 – 9]
= -20 / -5
= 4
Here, AB = BC = CA
∴ The Given points arecollinear.
(ii) A(16, – 18), B(3, –6), C(– 10, 6)
By using the formula,
The slope of the line= [y2 – y1] / [x2 – x1]
So,
The slope of line AB =[-6 – (-18)] / [3 – 16]
= 12 / -13
The slope of line BC =[6 – (-6)] / [-10 – 3]
= 12 / -13
The slope of line CA =[6 – (-18)] / [-10 – 16]
= 12 / -13
= 4
Here, AB = BC = CA
∴ The Given points arecollinear.