NCERT Solutions Class 11th Maths Chapter – 10 Straight Lines Exercise 10.1

NCERT Solutions Class 11th Maths Chapter – 10 Straight Lines Exercise 10.1

NCERT Solutions Class 11th Maths Chapter – 10 Straight Lines Exercise 10.1

?Chapter – 10?

✍Straight Lines✍

?Exercise 10.1?

1. Draw a quadrilateral in the Cartesian plane, whose vertices are (– 4, 5), (0, 7), (5, – 5) and (– 4, –2). Also, find its area.

?‍♂️solution – Let ABCD be the given quadrilateral with vertices A (-4,5), B (0,7), C (5.-5) and D (-4,-2).
Now let us plot the points on the Cartesian plane by joining the points AB, BC, CD, AD which gives us the required quadrilateral.

To find the area, draw diagonal AC
So, area (ABCD) = area (∆ABC) + area (∆ADC)
Then, area of triangle with vertices (x₁,y₁) , (x₂, y₂) and (x₃,y₃) is
Are of ∆ ABC = ½ [x₁ (y₂ – y₃) + x₂ (y₃ – y₁) + x₃ (y₁ – y₂)]
= ½ [-4 (7 + 5) + 0 (-5 – 5) + 5 (5 – 7)] unit²
= ½ [-4 (12) + 5 (-2)] unit²
= ½ (58) unit²
= 29 unit²
Are of ∆ ACD = ½ [x₁ (y₂ – y₃) + x₂ (y₃ – y₁) + x₃ (y₁ – y₂)]
= ½ [-4 (-5 + 2) + 5 (-2 – 5) + (-4) (5 – (-5))] unit²
= ½ [-4 (-3) + 5 (-7) – 4 (10)] unit²
= ½ (-63) unit²
= -63/2 unit²
Since area cannot be negative area ∆ ACD = 63/2 unit²
Area (ABCD) = 29 + 63/2
= 121/2 unit²

2. The base of an equilateral triangle with side 2a lies along the y-axis such that the mid-point of the base is at the origin. Find vertices of the triangle.

?‍♂️solution – 

Let us consider ABC be the given equilateral triangle with side 2a.
Where, AB = BC = AC = 2a
In the above figure, by assuming that the base BC lies on the x axis such that the mid-point of BC is at the origin i.e. BO = OC = a, where O is the origin.
The co-ordinates of point C are (0, a) and that of B are (0,-a)
Since the line joining a vertex of an equilateral ∆ with the mid-point of its opposite side is perpendicular.
So, vertex A lies on the y –axis
By applying Pythagoras theorem
(AC)² = OA² + OC²
(2a)²= a² + OC²
4a² – a² = OC²
3a² = OC²
OC =√3a
Co-ordinates of point C = ± √3a, 0
∴ The vertices of the given equilateral triangle are (0, a), (0, a), (√3a, 0)
Or (0, a), (0, -a) and (-√3a, 0)

3. Find the distance between P (x₁, y₁) and Q (x₂, y₂) when: (i) PQ is parallel to the y-axis, (ii) PQ is parallel to the x-axis.

?‍♂️solution – Given –
Points P (x₁, y₁) and Q(x₂, y₂)
(i) When PQ is parallel to y axis then  x₁ = x₂
So, the distance between P and Q is given by

= |y₂ – y₁|

(ii) When PQ is parallel to the x-axis then  y₁ = y₂
So, the distance between P and Q is given by =

= |x₂ – x₁|

4. Find a point on the x-axis, which is equidistant from the points (7, 6) and (3, 4).

?‍♂️solution – Let us consider (a, 0) be the point on the x-axis that is equidistant from the point (7, 6) and (3, 4).
So,

Now, let us square on both the sides we get,
a² – 14a + 85 = a² – 6a + 25
-8a = -60
a = 60/8
= 15/2
∴ The required point is (15/2, 0)

5. Find the slope of a line, which passes through the origin, and the mid-point of the line segment joining the points P (0, – 4) and B (8, 0).

?‍♂️solution – The co-ordinates of mid-point of the line segment joining the points P (0, – 4) and B (8, 0) are (0+8)/2, (-4+0)/2 = (4, -2)
The slope ‘m’ of the line non-vertical line passing through the point (x₁, y₁) and
(x₂, y₂) is given by m = (y₂ – y₁)/(x₂ – x₁) where, x ≠ x₁
The slope of the line passing through (0, 0) and (4, -2) is (-2 0)/(4-0) = -1/2
∴ The required slope is -1/2.

6. Without using the Pythagoras theorem, show that the points (4, 4), (3, 5) and (–1, –1) are the vertices of a right-angled triangle.

?‍♂️solution – The vertices of the given triangle are (4, 4), (3, 5) and (–1, –1).
The slope (m) of the line non-vertical line passing through the point (x₁, y₁) and
(x₂, y₂) is given by m = (y₂ – y₁)/(x₂ – x₁) where, x ≠ x₁
So, the slope of the line AB (m₁) = (5-4)/(3-4) = 1/-1 = -1
the slope of the line BC (m₂) = (-1-5)/(-1-3) = -6/-4 = 3/2
the slope of the line CA (m₃) = (4+1)/(4+1) = 5/5 = 1
It is observed that, m₁.m₃ = -1.1 = -1
Hence, the lines AB and CA are perpendicular to each other
∴ given triangle is right-angled at A (4, 4)
And the vertices of the right-angled ∆ are (4, 4), (3, 5) and (-1, -1)

7. Find the slope of the line, which makes an angle of 30° with the positive direction of y-axis measured anticlockwise.

?‍♂️solution – We know that, if a line makes an angle of 30° with the positive direction of y-axis measured anti-clock-wise , then the angle made by the line with the positive direction of x- axis measure anti-clock-wise is 90° + 30° = 120°
∴ The slope of the given line is tan 120° = tan (180° – 60°)
= – tan 60°
= –√3

8. Find the value of x for which the points (x, – 1), (2, 1) and (4, 5) are collinear.

?‍♂️solution – If the points (x, – 1), (2, 1) and (4, 5) are collinear, then Slope of AB = Slope of BC
Then, (1+1)/(2-x) = (5-1)/(4-2)
2/(2-x) = 4/2
2/(2-x) = 2
2 = 2(2-x)
2 = 4 – 2x
2x = 4 – 2
2x = 2
x = 2/2
= 1
∴ The required value of x is 1.

9. Without using distance formula, show that points (– 2, – 1), (4, 0), (3, 3) and (–3, 2) are the vertices of a parallelogram.

?‍♂️solution – 

Let the given point be A (-2, -1) , B (4, 0) , C ( 3, 3) and D ( -3, 2)
So now, The slope of AB = (0+1)/(4+2) = 1/6
The slope of CD = (3-2)/(3+3) = 1/6
Hence, slope of AB = Slope of CD
∴ AB ∥ CD
Now,
The slope of BC = (3-0)/(3-4) = 3/-1 = -3
The slope of AD = (2+1)/(-3+2) = 3/-1 = -3
Hence, slope of BC = Slope of AD
∴ BC ∥ AD
Thus the pair of opposite sides are quadrilateral are parallel,so
we can say that ABCD is a parallelogram.
Hence the given vertices, A (-2, -1), B (4, 0), C(3, 3) and D(-3, 2) are vertices of a parallelogram.

10. Find the angle between the x-axis and the line joining the points (3, –1) and (4, –2).

?‍♂️solution – The Slope of the line joining the points (3, -1) and (4, -2) is given by
m = (y₂ – y₁)/(x₂ – x₁) where, x ≠ x₁
m = (-2 –(-1))/(4-3)
= (-2+1)/(4-3)
= -1/1
= -1
The angle of inclination of line joining the points (3, -1) and (4, -2) is given by
tan θ = -1
θ = (90° + 45°) = 135°
∴ The angle between the x-axis and the line joining the points (3, –1) and (4, –2) is 135°.

11. The slope of a line is double of the slope of another line. If tangent of the angle between them is 1/3, find the slopes of the lines.

?‍♂️solution – Let us consider ‘m₁’ and ‘m’ be the slope of the two given lines such that m₁ = 2m
We know that if θ is the angle between the lines l1 and l2 with slope m1 and m₂, then

1+2m² = -3m
2m² +1 +3m = 0
2m (m+1) + 1(m+1) = 0
(2m+1) (m+1)= 0
m = -1 or -1/2
If m = -1, then the slope of the lines are -1 and -2
If m = -1/2, then the slope of the lines are -1/2 and -1
Case 2:

2m² – 3m + 1 = 0
2m² – 2m – m + 1 = 0
2m (m – 1) – 1(m – 1) = 0
m = 1 or 1/2
If m = 1, then the slope of the lines are 1 and 2
If m = 1/2, then the slope of the lines are 1/2 and 1
∴ The slope of the lines are [-1 and -2] or [-1/2 and -1] or [1 and 2] or [1/2 and 1]

12. A line passes through (x₁, y₁) and (h, k). If slope of the line is m, show that k – y₁ = m (h – x₁).

?‍♂️solution – Given: the slope of the line is ‘m’
The slope of the line passing through (x₁, y₁) and (h, k) is (k – y₁)/(h – x₁)
So,
(k – y₁)/(h – x₁)  = m
(k – y₁) = m (h – x₁)
Hence proved.

13. If three points (h, 0), (a, b) and (0, k) lie on a line, show that a/h + b/k = 1 

?‍♂️solution – Let us consider if the given points A (h, 0), B (a, b) and C (0, k) lie on a line
Then, slope of AB = slope of BC
(b – 0)/(a – h) = (k – b)/(0 – a)
let us simplify we get,
-ab = (k-b) (a-h)
-ab = ka- kh –ab +bh
ka +bh = kh
Divide both the sides by kh we get,
ka/kh + bh/kh = kh/kh
a/h + b/k = 1
Hence proved.

14. Consider the following population and year graph (Fig 10.10), find the slope of the line AB and using it, find what will be the population in the year 2010?

?‍♂️solution –  We know that, the line AB passes through points A (1985, 92) and B (1995, 97),
Its slope will be (97 – 92)/(1995 – 1985) = 5/10 = 1/2
Let ‘y’ be the population in the year 2010. Then, according to the given graph, AB must pass through point C (2010, y)
So now, slope of AB = slope of BC

15/2 = y – 97

y = 7.5 + 97 = 104.5

∴ The slope of the line AB is 1/2, while in the year 2010 the population will be 104.5 crores.