Commerce Exam  >  Commerce Notes  >  Mathematics (Maths) Class 11  >  NCERT Solutions: Exercise 9.2- Straight Lines

NCERT Solutions Class 11 Maths Chapter 9 - Straight Lines

Q1: Write the equations for the x and y-axes.
Ans: The y-coordinate of every point on the x-axis is 0.
Therefore, the equation of the x-axis is y = 0.
The x-coordinate of every point on the y-axis is 0.
Therefore, the equation of the y-axis is y = 0.

Q2: Find the equation of the line which passes through the point (–4, 3) with slopeNCERT Solutions Class 11 Maths Chapter 9 - Straight Lines .
Ans: We know that the equation of the line passing through point NCERT Solutions Class 11 Maths Chapter 9 - Straight Lines , whose slope is m, is NCERT Solutions Class 11 Maths Chapter 9 - Straight Lines .
Thus, the equation of the line passing through point (–4, 3), whose slope is NCERT Solutions Class 11 Maths Chapter 9 - Straight Lines , is

NCERT Solutions Class 11 Maths Chapter 9 - Straight Lines

Q3: Find the equation of the line which passes though (0, 0) with slope m.
Ans: We know that the equation of the line passing through point NCERT Solutions Class 11 Maths Chapter 9 - Straight Lines , whose slope is m, is NCERT Solutions Class 11 Maths Chapter 9 - Straight Lines .
Thus, the equation of the line passing through point (0, 0), whose slope is m,is
(y – 0) = m(x – 0)
i.e., y = mx

Q4: Find the equation of the line which passes though  NCERT Solutions Class 11 Maths Chapter 9 - Straight Lines and is inclined with the x-axis at an angle of 75°.
Ans: The slope of the line that inclines with the x-axis at an angle of 75° is m = tan 75°

NCERT Solutions Class 11 Maths Chapter 9 - Straight Lines
We know that the equation of the line passing through point NCERT Solutions Class 11 Maths Chapter 9 - Straight Lines , whose slope is m, is NCERT Solutions Class 11 Maths Chapter 9 - Straight Lines .
Thus, if a line passes though NCERT Solutions Class 11 Maths Chapter 9 - Straight Lines and inclines with the x-axis at an angle of 75°, then the equation of the line is given asNCERT Solutions Class 11 Maths Chapter 9 - Straight Lines

Q5: Find the equation of the line which intersects the x-axis at a distance of 3 units to the left of origin with slope –2.
Ans: It is known that if a line with slope m makes x-intercept d, then the equation of the line is given as
y = m(x – d)
For the line intersecting the x-axis at a distance of 3 units to the left of the origin, d = –3.
The slope of the line is given as m = –2
Thus, the required equation of the given line is
y = –2 [x – (–3)]
y = –2x – 6
i.e., 2x +  y + 6 = 0

Q6: Find the equation of the line which intersects the y-axis at a distance of 2 units above the origin and makes an angle of 30° with the positive direction of the x-axis.
Ans: It is known that if a line with slope m makes y-intercept c, then the equation of the line is given as
y = mx + c
Here, c = 2 and m = tan 30° NCERT Solutions Class 11 Maths Chapter 9 - Straight Lines .
Thus, the required equation of the given line is

NCERT Solutions Class 11 Maths Chapter 9 - Straight Lines

Q7: Find the equation of the line which passes through the points (–1, 1) and (2, –4).
Ans: t is known that the equation of the line passing through points (x1, y1) and (x2, y2) is NCERT Solutions Class 11 Maths Chapter 9 - Straight Lines .
Therefore, the equation of the line passing through the points (–1, 1) and
(2, –4) is

NCERT Solutions Class 11 Maths Chapter 9 - Straight Lines

Q8: The vertices of ΔPQR are P (2, 1), Q (–2, 3) and R (4, 5). Find equation of the median through the vertex R.
Ans: It is given that the vertices of ΔPQR are P (2, 1), Q (–2, 3), and R (4, 5).
Let RL be the median through vertex R.
Accordingly, L is the mid-point of PQ.
By mid-point formula, the coordinates of point L are given by NCERT Solutions Class 11 Maths Chapter 9 - Straight Lines
It is known that the equation of the line passing through points (x1, y1) and (x2, y2) is NCERT Solutions Class 11 Maths Chapter 9 - Straight Lines .
Therefore, the equation of RL can be determined by substituting (x1, y1) = (4, 5) and (x2, y2) = (0, 2).

NCERT Solutions Class 11 Maths Chapter 9 - Straight LinesHence, 

NCERT Solutions Class 11 Maths Chapter 9 - Straight Lines

NCERT Solutions Class 11 Maths Chapter 9 - Straight Lines
Thus, the required equation of the median through vertex R is NCERT Solutions Class 11 Maths Chapter 9 - Straight Lines .

Q9: Find the equation of the line passing through (–3, 5) and perpendicular to the line through the points (2, 5) and (–3, 6).
Ans: The slope of the line joining the points (2, 5) and (–3, 6) is NCERT Solutions Class 11 Maths Chapter 9 - Straight Lines
We know that two non-vertical lines are perpendicular to each other if and only if their slopes are negative reciprocals of each other.
Therefore, slope of the line perpendicular to the line through the points (2, 5) and (–3, 6)   NCERT Solutions Class 11 Maths Chapter 9 - Straight Lines
Now, the equation of the line passing through point (–3, 5), whose slope is 5, is

NCERT Solutions Class 11 Maths Chapter 9 - Straight Lines

Q10: A line perpendicular to the line segment joining the points (1, 0) and (2, 3) divides it in the ratio 1:n. Find the equation of the line.
Ans: According to the section formula, the coordinates of the point that divides the line segment joining the points (1, 0) and (2, 3) in the ratio 1: n is given by

NCERT Solutions Class 11 Maths Chapter 9 - Straight Lines
The slope of the line joining the points (1, 0) and (2, 3) is

NCERT Solutions Class 11 Maths Chapter 9 - Straight Lines
We know that two non-vertical lines are perpendicular to each other if and only if their slopes are negative reciprocals of each other.
Therefore, slope of the line that is perpendicular to the line joining the points (1, 0) and (2, 3)   NCERT Solutions Class 11 Maths Chapter 9 - Straight Lines
Now, the equation of the line passing through NCERT Solutions Class 11 Maths Chapter 9 - Straight Lines  and whose slope is NCERT Solutions Class 11 Maths Chapter 9 - Straight Lines is given by

NCERT Solutions Class 11 Maths Chapter 9 - Straight Lines

Q11: Find the equation of a line that cuts off equal intercepts on the coordinate axes and passes through the point (2, 3).
Ans: The equation of a line in the intercept form is

NCERT Solutions Class 11 Maths Chapter 9 - Straight Lines
Here, a and b are the intercepts on x and y axes respectively.
It is given that the line cuts off equal intercepts on both the axes. This means that a = b.
Accordingly, equation (i) reduces to

NCERT Solutions Class 11 Maths Chapter 9 - Straight Lines
Since the given line passes through point (2, 3), equation (ii) reduces to
2 + 3 = aa = 5
On substituting the value of a in equation (ii), we obtain
x  + y = 5, which is the required equation of the line

Q12: Find equation of the line passing through the point (2, 2) and cutting off intercepts on the axes whose sum is 9.
Ans: The equation of a line in the intercept form is

NCERT Solutions Class 11 Maths Chapter 9 - Straight Lines
Here, a and b are the intercepts on x and y axes respectively.
It is given that a  + b = 9 ⇒ b = 9 – a … (ii)
From equations (i) and (ii), we obtain

NCERT Solutions Class 11 Maths Chapter 9 - Straight Lines
It is given that the line passes through point (2, 2). Therefore, equation (iii) reduces to

NCERT Solutions Class 11 Maths Chapter 9 - Straight Lines
If a = 6 and b = 9 – 6 = 3, then the equation of the line is

NCERT Solutions Class 11 Maths Chapter 9 - Straight Lines
If a = 3 and b = 9 – 3 = 6, then the equation of the line is

NCERT Solutions Class 11 Maths Chapter 9 - Straight Lines

Q13: Find equation of the line through the point (0, 2) making an angle  NCERT Solutions Class 11 Maths Chapter 9 - Straight Lines with the positive x-axis. Also, find the equation of line parallel to it and crossing the y-axis at a distance of 2 units below the origin.
Ans: The slope of the line making an angle NCERT Solutions Class 11 Maths Chapter 9 - Straight Lines with the positive x-axis is  NCERT Solutions Class 11 Maths Chapter 9 - Straight Lines
Now, the equation of the line passing through point (0, 2) and having a slope  NCERT Solutions Class 11 Maths Chapter 9 - Straight Lines  is NCERT Solutions Class 11 Maths Chapter 9 - Straight Lines .

NCERT Solutions Class 11 Maths Chapter 9 - Straight Lines
The slope of line parallel to line NCERT Solutions Class 11 Maths Chapter 9 - Straight Lines  is NCERT Solutions Class 11 Maths Chapter 9 - Straight Lines .
It is given that the line parallel to line NCERT Solutions Class 11 Maths Chapter 9 - Straight Lines crosses the y-axis 2 units below the origin i.e., it passes through point (0, –2).
Hence, the equation of the line passing through point (0, –2) and having a slope NCERT Solutions Class 11 Maths Chapter 9 - Straight Lines  is

NCERT Solutions Class 11 Maths Chapter 9 - Straight Lines

Q14: The perpendicular from the origin to a line meets it at the point (– 2, 9), find the equation of the line.
Ans: The slope of the line joining the origin (0, 0) and point (–2, 9) is  NCERT Solutions Class 11 Maths Chapter 9 - Straight Lines
Accordingly, the slope of the line perpendicular to the line joining the origin and point (– 2, 9) is

NCERT Solutions Class 11 Maths Chapter 9 - Straight Lines
Now, the equation of the line passing through point (–2, 9) and having a slope m2 is

NCERT Solutions Class 11 Maths Chapter 9 - Straight Lines

Q15: The length L (in centimetre) of a copper rod is a linear function of its Celsius temperature C. In an experiment, if L = 124.942 when C = 20 and L = 125.134 when C = 110, express L in terms of C.
Ans: It is given that when C = 20, the value of L is 124.942, whereas when C = 110, the value of L is 125.134.
Accordingly, points (20, 124.942) and (110, 125.134) satisfy the linear relation between L and C.
Now, assuming C along the x-axis and L along the y-axis, we have two points i.e., (20, 124.942) and (110, 125.134) in the XY plane.
Therefore, the linear relation between L and C is the equation of the line passing through points (20, 124.942) and (110, 125.134).
(L – 124.942) =  NCERT Solutions Class 11 Maths Chapter 9 - Straight Lines

NCERT Solutions Class 11 Maths Chapter 9 - Straight Lines

Q16: The owner of a milk store finds that, he can sell 980 litres of milk each week at Rs 14/litre and 1220 litres of milk each week at Rs 16/litre. Assuming a linear relationship between selling price and demand, how many litres could he sell weekly at Rs 17/litre?
Ans: The relationship between selling price and demand is linear.
Assuming selling price per litre along the x-axis and demand along the y-axis, we have two points i.e., (14, 980) and (16, 1220) in the XY plane that satisfy the linear relationship between selling price and demand.
Therefore, the linear relationship between selling price per litre and demand is the equation of the line passing through points (14, 980) and (16, 1220).

NCERT Solutions Class 11 Maths Chapter 9 - Straight Lines

NCERT Solutions Class 11 Maths Chapter 9 - Straight Lines

When x = Rs 17/litre,

NCERT Solutions Class 11 Maths Chapter 9 - Straight Lines

Thus, the owner of the milk store could sell 1340 litres of milk weekly at Rs 17/litre.

Q17: P (a, b) is the mid-point of a line segment between axes. Show that equation of the line is  NCERT Solutions Class 11 Maths Chapter 9 - Straight Lines
Ans: Let AB be the line segment between the axes and let P (a, b) be its mid-point.

NCERT Solutions Class 11 Maths Chapter 9 - Straight LinesLet the coordinates of A and B be (0, y) and (x, 0) respectively.
Since P (a, b) is the mid-point of AB,

NCERT Solutions Class 11 Maths Chapter 9 - Straight Lines

Thus, the respective coordinates of A and B are (0, 2b) and (2a, 0).
The equation of the line passing through points (0, 2b) and (2a, 0) is

NCERT Solutions Class 11 Maths Chapter 9 - Straight Lines
On dividing both sides by ab, we obtain

NCERT Solutions Class 11 Maths Chapter 9 - Straight Lines
Thus,the equation of the line is NCERT Solutions Class 11 Maths Chapter 9 - Straight Lines .

Q18: Point R (h, k) divides a line segment between the axes in the ratio 1:2. Find equation of the line.
Ans: Let AB be the line segment between the axes such that point R (h, k) divides AB in the ratio 1: 2.
Let the respective coordinates of A and B be (x, 0) and (0, y).
Since point R (h, k) divides AB in the ratio 1: 2, according to the section formula,

NCERT Solutions Class 11 Maths Chapter 9 - Straight Lines

NCERT Solutions Class 11 Maths Chapter 9 - Straight Lines
Therefore, the respective coordinates of A and B are NCERT Solutions Class 11 Maths Chapter 9 - Straight Lines  and (0, 3k).
Now, the equation of line AB passing through points  NCERT Solutions Class 11 Maths Chapter 9 - Straight Lines  and (0, 3k) is

NCERT Solutions Class 11 Maths Chapter 9 - Straight Lines
Thus, the required equation of the line is 2kx + hy = 3hk.

Q19: By using the concept of equation of a line, prove that the three points (3, 0), (–2, –2) and (8, 2) are collinear.
Ans: In order to show that points (3, 0), (–2, –2), and (8, 2) are collinear, it suffices to show that the line passing through points (3, 0) and (–2, –2) also passes through point (8, 2).
The equation of the line passing through points (3, 0) and (–2, –2) is

NCERT Solutions Class 11 Maths Chapter 9 - Straight Lines
It is observed that at x = 8 and y = 2,
L.H.S. = 2 × 8 – 5 × 2 = 16 – 10 = 6 = R.H.S.
Therefore, the line passing through points (3, 0) and (–2, –2) also passes through point (8, 2). Hence, points (3, 0), (–2, –2), and (8, 2) are collinear.

The document NCERT Solutions Class 11 Maths Chapter 9 - Straight Lines is a part of the Commerce Course Mathematics (Maths) Class 11.
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FAQs on NCERT Solutions Class 11 Maths Chapter 9 - Straight Lines

1. What is the equation of a straight line in the slope-intercept form?
Ans. The equation of a straight line in the slope-intercept form is given by y = mx + c, where m is the slope of the line and c is the y-intercept.
2. How can I find the slope of a straight line passing through two given points?
Ans. To find the slope of a straight line passing through two given points (x1, y1) and (x2, y2), we can use the formula m = (y2 - y1) / (x2 - x1), where m represents the slope.
3. What is the relation between the slopes of two perpendicular lines?
Ans. The slopes of two perpendicular lines are negative reciprocals of each other. In other words, if the slope of one line is m, then the slope of the perpendicular line will be -1/m.
4. Can a straight line have two y-intercepts?
Ans. No, a straight line can have only one y-intercept. The y-intercept is the point where the line intersects the y-axis, and it is unique for any given line.
5. How do I determine if two lines are parallel or not?
Ans. Two lines are parallel if and only if their slopes are equal. If the slopes of two lines are different, then they are not parallel.
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