(a) Find the points which divide the line joining A(1,3) and B(-4,-1) internally in the ratios:

(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).

ii) half way: (-5/2,2)

iii) 3/4 the way: (-5*3/4, 1)

To find the points that divide the line joining A(1,3) and B(-4,-1) internally in the given ratios, we can use the section formula. The section formula states that if we have two points A(x1, y1) and B(x2, y2), and we want to find a point P that divides the line AB in the ratio m:n, then the coordinates of point P can be found using the formulas:

Px = (mx2 + nx1) / (m + n)
Py = (my2 + ny1) / (m + n)

(a)
(ii) Ratio 1:1
This means we are dividing the line AB into two equal parts. Since the ratio is 1:1, both m and n are equal to 1. Using the section formula, we can calculate the coordinates of the point P:

Px = (1*(-4) + 1*1) / (1 + 1) = (-4 + 1) / 2 = -3/2
Py = (1*(-1) + 1*3) / (1 + 1) = (-1 + 3) / 2 = 1

So, the point P that divides the line AB internally in the ratio 1:1 is P(-3/2, 1).

(iii) Ratio 3:1
This means we are dividing the line AB into four parts, with three parts on the side of point A and one part on the side of point B. Using the section formula, we can calculate the coordinates of the point P:

Px = (3*(-4) + 1*1) / (3 + 1) = (-12 + 1) / 4 = -11/4
Py = (3*(-1) + 1*3) / (3 + 1) = (-3 + 3) / 4 = 0

So, the point P that divides the line AB internally in the ratio 3:1 is P(-11/4, 0).

(b)
To plot the points and draw the line AB on a graph, we can use the given coordinates A(1,3) and B(-4,-1). The x-axis represents the horizontal axis, and the y-axis represents the vertical axis.

1. Plot Point A: A(1,3)
- Locate the point (1,3) on the graph by moving 1 unit along the x-axis and 3 units up along the y-axis. Mark this point as A.

2. Plot Point B: B(-4,-1)
- Locate the point (-4,-1) on the graph by moving 4 units to the left along the x-axis and 1 unit down along the y-axis. Mark this point as B.

3. Draw the Line AB:
- Connect points A and B using a straight line. This line represents the line joining A and B.

4. Plot the calculated points:
- Plot the point P(−3/2, 1) on the graph. Locate this point by moving -3/2 units along the x-axis and 1 unit up along the y-axis. Mark this point as P.
- Plot the point P(−11/4, 0) on the graph. Locate this point by moving -11/4 units along the x-axis. Since the y-coordinate is 0, this means it lies on the x-axis. Mark this point as Q.

Now, you should have points A, B, P, and Q plotted on the graph, with the line AB passing through points A and B.