Question 21
Question: Find the equation of the line through the intersection of the lines
6x+5y=11, 8x-5y=3 and parallel to the line 3x - 2y + 12 = 0.
Answer: The given lines are
6x + 5y = 11 ...(i)
8x - 5y = 3 ...(ii)
Solving for x and y, the coordinates of the point of intersection of the two lines are (1,1).
The slope of the line through (1,1) = Slope of line 3x - 2y + 12 = 0
\ The equation of the line which passes through (1, 1) and whose slope is 3/2 is
Question 22
Question: Show that the points (1,4) and (-1,-2) lie on either side of line 4x + 5y + 10 = 0.
Answer: Let d = distance between the line 4x + 5y + 10 = 0 and the point (1,4).
Let d' be the distance between the line 4x + 5y + 10 = 0 and the point (-1,-2).
Since d and d' are of the opposite signs, the points lie on the either side of the line.
Question 23
Question: Find the distance of the line 24x + 7y + 12= 0 from the point (4,6).
Answer: Line 24x + 7y + 12 = 0, point (4,6).
6 units (since distance cannot be negative)
Question 24
Question: Find the distance between the following pairs of parallel lines.
i) 3x + 4y + 20 = 0, 6x + 8y + 45 =0
ii) 12x + 15y + 41 = 0, 4x + 5y + 41 = 0
Answer: i) 3x + 4y + 20 = 0, 6x + 8y + 45 = 0
Two lines are 3x + 4y + 20 = 0 ...(i)
6x + 8y + 45 = 0 ...(ii)
Distance of (i) from the origin (0,0) is
Distance of (ii) from the origin (0,0) is
Distance between the lines is
ii) 12x + 15y + 41 = 0, 4x + 5y + 41 = 0
The lines are 12x + 15y + 41 = 0 ...(i)
4x + 5y + 41 = 0 ...(ii)
The distance of line (i) from (0,0) is
The distance of line (ii) from (0,0) is
Distance between the lines is
Question 25
Question: Find the point of intersection of the lines
Answer: 
2x + 3y - 7 = 0
3x - 5y - 1 = 0
The point of intersection is (2,1).
ii) mx - y - c = 0
x - my + c = 0
Adding (i) and (ii), we get
Subtracting (ii) from (i), we get
2mx = 0
x = 0
Question 26
Question: Find the centroid of the triangle whose sides are
x + y - 1 = 0, x - 3y + 3 = 0 and x - y - 1 = 0.
Answer: Let AB represent the side x + y - 1 = 0 ...(i)
Let BC represent the side x - 3y + 3 = 0 ... (ii)
Let CA represent the side x - y - 1 = 0 ... (iii)
Solving (i) and (ii) for x and y, we get the coordinates of B, i.e., B(0,1).
Solving (ii) and (iii) , we get the coordinates of C, i.e., C(3,2).
Solving (iii) and (i), we get the coordinates of A, i.e., A(1,0).
The coordinates of the centroid of the triangle are
