NCERT Solutions Class 11th Maths Chapter – 10 Straight Lines Miscellaneous Exercise

NCERT Solutions Class 11th Maths Chapter – 10 Straight Lines Miscellaneous Exercise

NCERT Solutions Class 11th Maths Chapter – 10 Straight Lines Miscellaneous Exercise

?Chapter – 10?

✍Straight Lines✍

?Miscellaneous Exercise?

1. Find the values of k for which the line (k – 3) x – (4 – k²) y + k² – 7k + 6 = 0 is

(a) Parallel to the x-axis,

(b) Parallel to the y-axis,

(c) Passing through the origin.

?‍♂️solution – It is given that
(k – 3) x – (4 – k²) y + k² – 7k + 6 = 0 … (1)
(a) Here if the line is parallel to the x-axis
Slope of the line = Slope of the x-axis
It can be written as
(4 – k²) y = (k – 3) x + k² – 7k + 6 = 0
We get

By further calculation
k – 3 = 0
k = 3
Hence, if the given line is parallel to the x-axis, then the value of k is 3.
(b) Here if the line is parallel to the y-axis, it is vertical and the slope will be undefined.
So the slope of the given line

k² = 4
k = ± 2
Hence, if the given line is parallel to the y-axis, then the value of k is ± 2.
(c) Here if the line is passing through (0, 0) which is the origin satisfies the given equation of line.
(k – 3) (0) – (4 – k²) (0) +k² – 7k + 6 = 0
By further calculation
k² – 7k + 6 = 0
Separating the terms
k² – 6k – k + 6 = 0
We get
(k – 6) (k – 1) = 0
k = 1 or 6
Hence, if the given line is passing through the origin, then the value of k is either 1 or 6.

2. Find the values of θ and p, if the equation x cos θ + y sin θ = p is the normal form of the line √3x + y + 2 = 0.

?‍♂️solution – 

3. Find the equations of the lines, which cut-off intercepts on the axes whose sum and product are 1 and –6, respectively.

?‍♂️solution – Consider the intercepts cut by the given lines on a and b axes.
a + b = 1 …… (1)
ab = – 6 …….. (2)
By solving both the equations we get
a = 3 and b = -2 or a = – 2 and b = 3
We know that the equation of the line whose intercepts on a and b axes is

Case I – a = 3 and b = – 2
So the equation of the line is – 2x + 3y + 6 = 0, i.e. 2x – 3y = 6.
Case II – a = -2 and b = 3
So the equation of the line is 3x – 2y + 6 = 0, i.e. -3x + 2y = 6
Hence, the required equation of the lines are 2x – 3y = 6 and -3x + 2y = 6.

4. What are the points on the y-axis whose distance from the line x/3 + y/4 = 1 is 4 units.

?‍♂️solution – Consider (0,b) as the point on the y-axis whose distance from line x/3 + y/4 = 1 is 4 units.
It can be written as 4x + 3y – 12 = 0 ……. (1)
By comparing equation (1) to the general equation of line Ax + By + C = 0, we get
A = 4, B = 3 and C = – 12
We know that the perpendicular distance (d) of a line Ax + By + C = 0 from (x₁, y₁) is written as

By cross multiplication
20 = |3b – 12|
We get
20 = ± (3b – 12)
Here 20 = (3b – 12) or 20 = – (3b – 12)
It can be written as
3b = 20 + 12 or 3b = -20 + 12
So we get
b = 32/3 or b = -8/3
Hence, the required points are (0, 32/3) and (0, -8/3).

5. Find the perpendicular distance from the origin to the line joining the points

?‍♂️solution – 

6. Find the equation of the line parallel to y-axis and drawn through the point of intersection of the lines x – 7y + 5 = 0 and 3x + y = 0.

?‍♂️solution – Here the equation of any line parallel to the y-axis is of the form
x = a ……. (1)
Two given lines are
x – 7y + 5 = 0 …… (2)
3x + y = 0 …… (3)
By solving equations (2) and (3) we get
x = -5/22 and y = 15/22
(-5/ 22, 15/22) is the point of intersection of lines (2) and (3)
If the line x = a passes through point (-5/22, 15/22) we get a = -5/22
Hence, the required equation of the line is x = -5/22.

7. Find the equation of a line drawn perpendicular to the line x/4 + y/6 = 1 through the point, where it meets the y-axis.

?‍♂️solution – It is given that
x/4 + y/6 = 1
We can write it as
3x + 2y – 12 = 0
So we get
y = -3/2 x + 6, which is of the form y = mx + c
Here the slope of the given line = -3/2
So the slope of line perpendicular to the given line = -1/ (-3/2) = 2/3
Consider the given line intersect the y-axis at (0, y)
By substituting x as zero in the equation of the given line
y/6 = 1
y = 6
Hence, the given line intersects the y-axis at (0, 6)
We know that the equation of the line that has a slope of 2/3 and passes through point (0, 6) is
(y – 6) = 2/3 (x – 0)
By further calculation
3y – 18 = 2x
So we get
2x – 3y + 18 = 0
Hence, the required equation of the line is 2x – 3y + 18 = 0.

8. Find the area of the triangle formed by the lines y – x = 0, x + y = 0 and x – k = 0.

?‍♂️solution – It is given that
y – x = 0 …… (1)
x + y = 0 …… (2)
x – k = 0 ……. (3)
Here the point of intersection of
Lines (1) and (2) is
x = 0 and y = 0
Lines (2) and (3) is
x = k and y = – k
Lines (3) and (1) is
x = k and y = k
So the vertices of the triangle formed by the three given lines are (0, 0), (k, -k) and (k, k)
Here the area of triangle whose vertices are (x₁, y₁), (x₂, y₂) and (x₃, y₃) is
½ |x₁ (y₂ – y₃) + x₂ (y₃ – y₁) + x₃ (y₁ – y₂)|
So the area of triangle formed by the three given lines
= ½ |0 (-k – k) + k (k – 0) + k (0 + k)| square units
By further calculation
= ½ |k² + k²| square units
So we get
= ½ |2k²|
= k² square units

9. Find the value of p so that the three lines 3x + y – 2 = 0, px + 2y – 3 = 0 and 2x – y – 3 = 0 may intersect at one point.

?‍♂️solution – It is given that
3x + y – 2 = 0 …… (1)
px + 2y – 3 = 0 ….. (2)
2x – y – 3 = 0 …… (3)
By solving equations (1) and (3) we get
x = 1 and y = -1
Here the three lines intersect at one point and the point of intersection of lines (1) and (3) will also satisfy line (2)
p (1) + 2 (-1) – 3 = 0
By further calculation
p – 2 – 3 = 0
So we get
p = 5
Hence, the required value of p is 5.

10. If three lines whose equations are y = m₁x + c₁, y = m₂x + c₂ and y = m₃x + c₃ are concurrent, then show that m₁ (c₂ – c₃) + m₂ (c₃ – c₁) + m₃ (c₁ – c₂) = 0.

?‍♂️solution – It is given that
y = m₁x + c₁ ….. (1)
y = m₂x + c₂ ….. (2)
y = m₃x + c₃ ….. (3)
By subtracting equation (1) from (2) we get
0 = (m₂ – m₁) x + (c₂ – c₁)
(m₁ – m₂) x = c₂ – c₁
So we get

Taking out the common terms
m₁ (c₂ – c₃) + m₂ (c₃ – c₁) + m₃ (c₁ – c₂) = 0
Therefore, m₁ (c₂ – c₃) + m₂ (c₃ – c₁) + m₃ (c₁ – c₂) = 0.

11. Find the equation of the lines through the point (3, 2) which make an angle of 45° with the line x –2y = 3.

?‍♂️solution – Consider m1 as the slope of the required line
It can be written as
y = 1/2 x – 3/2 which is of the form y = mx + c
So the slope of the given line m2 = 1/2
We know that the angle between the required line and line x – 2y = 3 is 45o
If θ is the acute angle between lines l1 and l2 with slopes m1 and m2

It can be written as
2 + m₁ = 1 – 2m₁ or 2 + m₁ = – 1 + 2m₁
m₁ = – 1/3 or m₁ = 3
Case I – m₁ = 3
Here the equation of the line passing through (3, 2) and having a slope 3 is
y – 2 = 3 (x – 3)
By further calculation
y – 2 = 3x – 9
So we get
3x – y = 7
Case II – m1 = -1/3
Here the equation of the line passing through (3, 2) and having a slope -1/3 is
y – 2 = – 1/3 (x – 3)
By further calculation
3y – 6 = – x + 3
So we get
x + 3y = 9
Hence, the equations of the lines are 3x – y = 7 and x + 3y = 9.

12. Find the equation of the line passing through the point of intersection of the lines 4x + 7y – 3 = 0 and 2x – 3y + 1 = 0 that has equal intercepts on the axes.

?‍♂️solution – Consider the equation of the line having equal intercepts on the axes as
x/a + y/a = 1
It can be written as
x + y = a ….. (1)
By solving equations 4x + 7y – 3 = 0 and 2x – 3y + 1 = 0 we get
x = 1/13 and y = 5/13
(1/13, 5/13) is the point of intersection of two given lines
We know that equation (1) passes through point (1/13, 5/13)
1/13 + 5/13 = a
a = 6/13
So the equation (1) passes through (1/13, 5/13)
1/13 + 5/13 = a
We get
a = 6/13
Here the equation (1) becomes
x + y = 6/13
13x + 13y = 6
Hence, the required equation of the line is 13x + 13y = 6.

13. Show that the equation of the line passing through the origin and making an angle θ with the line y = mx + c is

?‍♂️solution – Consider y = m₁x as the equation of the line passing through the origin

14. In what ratio, the line joining (–1, 1) and (5, 7) is divided by the line x + y = 4?

?‍♂️solution – 

By cross multiplication
– k + 5 = 1 + k
We get
2k = 4
k = 2
Hence, the line joining the points (-1, 1) and (5, 7) is divided by the line x + y = 4 in the ratio 1: 2.

15. Find the distance of the line 4x + 7y + 5 = 0 from the point (1, 2) along the line 2x – y = 0.

?‍♂️solution – It is given that
2x – y = 0 ….. (1)
4x + 7y + 5 = 0 …… (2)
Here A (1, 2) is a point on the line (1)
Consider B as the point of intersection of lines (1) and (2)

By solving equations (1) and (2) we get x = -5/18 and y = – 5/9
So the coordinates of point B are (-5/18, -5/9)
From distance formula the distance between A and B

16. Find the direction in which a straight line must be drawn through the point (–1, 2) so that its point of intersection with the line x + y = 4 may be at a distance of 3 units from this point.

?‍♂️solution – Consider y = mx + c as the line passing through the point (-1, 2)
So we get
2 = m (-1) + c
By further calculation
2 = -m + c
c = m + 2
Substituting the value of c
y = mx + m + 2 …… (1)
So the given line is
x + y = 4 ……. (2)
By solving both the equations we get

By cross multiplication
1 + m² = m² + 1 + 2m
So we get
2m = 0
m = 0
Hence, the slope of the required line must be zero i.e. the line must be parallel to the x-axis.

17. The hypotenuse of a right angled triangle has its ends at the points (1, 3) and (−4, 1). Find the equation of the legs (perpendicular sides) of the triangle.

?‍♂️solution – Consider ABC as the right angles triangle where ∠C = 90o
Here infinity such lines are present.
m is the slope of AC
So the slope of BC = -1/m
Equation of AC –
y – 3 = m (x – 1)
By cross multiplication
x – 1 = 1/m (y – 3)
Equation of BC –
y – 1 = – 1/m (x + 4)
By cross multiplication
x + 4 = – m (y – 1)
By considering values of m we get
If m = 0,
So we get
y – 3 = 0, x + 4 = 0
If m = ∞,
So we get
x – 1 = 0, y – 1 = 0 we get x = 1, y = 1

18. Find the image of the point (3, 8) with respect to the line x + 3y = 7 assuming the line to be a plane mirror.

?‍♂️solution – It is given that
x + 3y = 7 ….. (1)
Consider B (a, b) as the image of point A (3, 8)
So line (1) is perpendicular bisector of AB.

On further simplification
a + 3b = – 13 ….. (3)
By solving equations (2) and (3) we get
a = – 1 and b = – 4
Hence, the image of the given point with respect to the given line is (-1, -4).

19. If the lines y = 3x + 1 and 2y = x + 3 are equally inclined to the line y = mx + 4, find the value of m.

?‍♂️solution – It is given that
y = 3x + 1 …… (1)
2y = x + 3 …… (2)
y = mx + 4 …… (3)
Here the slopes of
Line (1), m₁ = 3
Line (2), m₂ = ½
Line (3), m₃ = m
We know that the lines (1) and (2) are equally inclined to line (3) which means that the angle between lines (1) and (3) equals the angle between lines (2) and (3).

On further calculation
– m² + m + 6 = 1 + m – 6m²
So we get
5m² + 5 = 0
Dividing the equation by 5
m² + 1 = 0
m = √-1, which is not real.
Therefore, this case is not possible.
If

20. If sum of the perpendicular distances of a variable point P (x, y) from the lines x + y – 5 = 0 and 3x – 2y + 7 = 0 is always 10. Show that P must move on a line.

?‍♂️solution – 

In the same way we can find the equation of line for any signs of (x + y – 5) and (3x – 2y + 7)
Hence, point P must move on a line.

21. Find equation of the line which is equidistant from parallel lines 9x + 6y – 7 = 0 and 3x + 2y + 6 = 0.

?‍♂️solution – 

Here
9h + 6k – 7 = 3 (3h + 2k + 6) or 9h + 6k – 7 = – 3 (3h + 2k + 6)
9h + 6k – 7 = 3 (3h + 2k + 6) is not possible as
9h + 6k – 7 = 3 (3h + 2k + 6)
By further calculation
– 7 = 18 (which is wrong)
We know that
9h + 6k – 7 = -3 (3h + 2k + 6)
By multiplication
9h + 6k – 7 = -9h – 6k – 18
We get
18h + 12k + 11 = 0
Hence, the required equation of the line is 18x + 12y + 11 = 0.

22. A ray of light passing through the point (1, 2) reflects on the x-axis at point A and the reflected ray passes through the point (5, 3). Find the coordinates of A.

?‍♂️solution – 

Consider the coordinates of point A as (a, 0)
Construct a line (AL) which is perpendicular to the x-axis
Here the angle of incidence is equal to angle of reflection
∠BAL = ∠CAL = Φ
∠CAX = θ
It can be written as
∠OAB = 180° – (θ + 2Φ) = 180° – [θ + 2(90° – θ)]
On further calculation
= 180° – θ – 180° + 2θ
= θ
So we get
∠BAX = 180° – θ

By cross multiplication
3a – 3 = 10 – 2a
We get
a = 13/5
Hence, the coordinates of point A are (13/5, 0).

23. Prove that the product of the lengths of the perpendiculars drawn from the points to the line

?‍♂️solution – It is given that

We can write it as
bx cos θ + ay sin θ – ab = 0 ….. (1)

24. A person standing at the junction (crossing) of two straight paths represented by the equations 2x – 3y + 4 = 0 and 3x + 4y – 5 = 0 wants to reach the path whose equation is 6x – 7y + 8 = 0 in the least time. Find equation of the path that he should follow.

?‍♂️solution – It is given that
2x – 3y + 4 = 0 …… (1)
3x + 4y – 5 = 0 ……. (2)
6x – 7y + 8 = 0 …… (3)
Here the person is standing at the junction of the paths represented by lines (1) and (2).
By solving equations (1) and (2) we get
x = – 1/17 and y = 22/17
Hence, the person is standing at point (-1/17, 22/17).
We know that the person can reach path (3) in the least time if he walks along the perpendicular line to (3) from point (-1/17, 22/17)
Here the slope of the line (3) = 6/7
We get the slope of the line perpendicular to line (3) = -1/ (6/7) = – 7/6
So the equation of line passing through (-1/17, 22/17) and having a slope of -7/6 is written as

By further calculation
6 (17y – 22) = – 7 (17x + 1)
By multiplication
102y – 132 = – 119x – 7
We get
1119x + 102y = 125
Therefore, the path that the person should follow is 119x + 102y = 125.