Chapter 10 Straight lines Ex 10.1 Solutions
Question - 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.
Answer - 11 : -
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 withslope 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 linesare -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 linesare 1/2 and 1
∴ Theslope of the lines are [-1 and -2] or [-1/2 and -1] or [1 and 2]or [1/2 and 1]
Question - 12 : - A line passesthrough (x1, y1) and (h, k). If slope ofthe line is m, show that k – y1 =m (h – x1).
Answer - 12 : -
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.
Question - 13 : - If three points (h, 0), (a, b) and (0, k) lie on a line, show that a/h + b/k = 1
Answer - 13 : -
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.
Question - 14 : - Consider the given population and year graph. Find the slope of the line AB and using it, find what will be the population in the year 2010?
Answer - 14 : -
Since line AB passes through points A (1985, 92) and B (1995, 97), its slope is
Let y be the population in the year 2010. Then, according to the given graph, line AB must pass through point C (2010, y).
∴Slope of AB = Slope of BC
Thus, the slope of line AB is
, while in the year 2010, the population will be 104.5 crores.