(a) Find the points which divide the line joining A(1,3) and B(-4,-1) internally in the ratios:
(i) 1:2
(ii) 1:1
(iii) 3:1
(b) On the same x-y axes plot the point A(1,3) and B(-4,-1) and draw the line joining them. Add the three points which you calculated in part (a).
(1,3), (x,y), (-4,-1).
(i). -4-x = 2(x-1).
-4-x = 2x-2, X = -2/3.
-1-y = 2(y-3).
-1-y = 2y-6, Y = 5/3.
(ii). x-1 = -4-x.
X = -3/2.
y-3 = -1-y. Y = 1.
(iii). x-1 = 3(-4-x).
x-1 = -12-3x, X = -11/4.
y-3 = 3(-1-y).
y-3 = -3-3y, Y = 0.
To find the points which divide the line joining A(1,3) and B(-4,-1) internally in the given ratios, we can use the section formula.
(a) Section Formula:
(i) For the ratio 1:2, let's denote the coordinates of the point P as (x,y). According to the section formula, the coordinates of P can be calculated as follows:
x = (2 * x2 + 1 * x1) / (1 + 2)
y = (2 * y2 + 1 * y1) / (1 + 2)
where (x1, y1) = A(1,3) and (x2, y2) = B(-4,-1).
Substituting the values, we get:
x = (2 * (-4) + 1 * 1) / (1 + 2)
y = (2 * (-1) + 1 * 3) / (1 + 2)
Simplifying this:
x = (-8 + 1) / 3
y = (-2 + 3) / 3
So, the coordinates of the point dividing the line AB in the ratio 1:2 internally are:
P = (-7/3, 1/3)
(ii) For the ratio 1:1, the section formula simplifies to finding the midpoint of AB.
x = (x1 + x2) / 2
y = (y1 + y2) / 2
Substituting the values:
x = (1 + (-4)) / 2
y = (3 + (-1)) / 2
Simplifying:
x = -3/2
y = 1/2
So, the coordinates of the midpoint of AB are:
M = (-3/2, 1/2)
(iii) For the ratio 3:1, we can use the same formula as in (i) but change the ratios:
x = (1 * x2 + 3 * x1) / (1 + 3)
y = (1 * y2 + 3 * y1) / (1 + 3)
Substituting the values:
x = (1 * (-4) + 3 * 1) / (1 + 3)
y = (1 * (-1) + 3 * 3) / (1 + 3)
Simplifying:
x = (-4 + 3) / 4
y = (-1 + 9) / 4
So, the coordinates of the point dividing the line AB in the ratio 3:1 internally are:
Q = (-1/4, 2)
(b) To plot the points A(1,3) and B(-4,-1) and draw the line joining them, you can use a graphing tool or simply graph paper and a ruler. Follow these steps:
1. Draw the x-y coordinate axes on your graph paper.
2. Plot point A(1,3) by moving 1 unit to the right on the x-axis and 3 units up on the y-axis. Mark this point.
3. Plot point B(-4,-1) by moving 4 units to the left on the x-axis and 1 unit down on the y-axis. Mark this point.
4. Connect points A and B to form a line segment.
5. Label the line segment as AB.
6. Plot the three points calculated in part (a):
- Point P = (-7/3, 1/3) for the ratio 1:2
- Point M = (-3/2, 1/2) for the ratio 1:1
- Point Q = (-1/4, 2) for the ratio 3:1
7. Label these points accordingly.
Now, you have the points A(1,3) and B(-4,-1) along with the three points P, M, and Q plotted on the same x-y axes, with the line AB connecting points A and B.