Chapter 10 Straight lines Ex 10.2 Solutions
Question - 11 : - 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.
Answer - 11 : -
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
The slope of the line joining the points (1, 0) and (2, 3) is
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)
Now, the equation of the line passing through
and whose slope is 
is given by
Question - 12 : - Find the equation of a line that cuts off equal intercepts on the coordinate axes and passes through the point (2, 3).
Answer - 12 : -
The equation of a line in the intercept form is
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
Since the given line passes through point (2, 3), equation (ii) reduces to
2 + 3 = a ⇒ a = 5
On substituting the value of a in equation (ii), we obtain
x + y = 5, which is the required equation of the line
Question - 13 : - Find equation of the line passing through the point (2, 2) and cutting off intercepts on the axes whose sum is 9.
Answer - 13 : -
The equation of a line in the intercept form is
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
It is given that the line passes through point (2, 2). Therefore, equation (iii) reduces to
If a = 6 and b = 9 – 6 = 3, then the equation of the line is
If a = 3 and b = 9 – 3 = 6, then the equation of the line is
Question - 14 : - Find equation of the line through the point (0, 2) making an angle 
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.
Answer - 14 : -
The slope of the line making an angle
with the positive x-axis is 
Now, the equation of the line passing through point (0, 2) and having a slope
is 
.
The slope of line parallel to line
is .It is given that the line parallel to line
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
is
Question - 15 : - The perpendicular from the origin to a line meets it at the point (– 2, 9), find the equation of the line.
Answer - 15 : -
The slope of the line joining the origin (0, 0) and point (–2, 9) is
Accordingly, the slope of the line perpendicular to the line joining the origin and point (– 2, 9) is
Now, the equation of the line passing through point (–2, 9) and having a slope m2 is
Question - 16 : - 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.
Answer - 16 : -
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) =
Answer - 17 : -
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).
When x = Rs 17/litre,
Thus, the owner of the milk store could sell 1340 litres of milk weekly at Rs 17/litre.
Question - 18 : - P (a, b) is the mid-point of a line segment between axes. Show that equation of the line is 
Answer - 18 : -
Let AB be the line segment between the axes and let P (a, b) be its mid-point.
Let the coordinates of A and B be (0, y) and (x, 0) respectively.
Since P (a, b) is the mid-point of AB,
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
On dividing both sides by ab, we obtain
Thus,the equation of the line is
.
Question - 19 : - Point R (h, k) divides a line segment between the axes in the ratio 1:2. Find equation of the line.
Answer - 19 : - 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,
Therefore, the respective coordinates of A and B are
and (0, 3k).Now, the equation of line AB passing through points
and (0, 3k) isThus,the required equation of the line is 2kx + hy = 3hk.
Question - 20 : - By using the concept of equation of a line, prove that the three points (3, 0), (–2, –2) and (8, 2) are collinear.
Answer - 20 : -
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
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.