(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).
I will do the first one, you do the other two in the same way.
Let P(x, y) be on AB so that
AP/PB = 1/2
for the x:
(x-1)/(-4-x) = 1/2
2x - 2 = -4 - x
3x = -2
x = -2/3
for the y:
(y-3)/(-1-y) = 1/2
2y - 6 = -1 - y
3y = 5
y = 5/3
so P is (-2/3 , 5/3)
for b), I am sure you meant "plot" the three points.
We cannot "add" points in this context.
(a) 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.
(i) Ratio 1:2
Let the point dividing the line internally in the ratio 1:2 be P.
Using the section formula, the coordinates of P(x,y) can be calculated as:
x = (2 * x2 + 1 * x1) / (1 + 2)
y = (2 * y2 + 1 * y1) / (1 + 2)
Substituting the coordinates of A(1,3) and B(-4,-1), we find:
x = (2 * (-4) + 1 * 1) / 3
y = (2 * (-1) + 1 * 3) / 3
Simplifying, we get:
x = ( -8 + 1) / 3 = -7/3
y = ( -2 + 3) / 3 = 1/3
So, the point which divides the line joining A and B internally in the ratio 1:2 is P(-7/3, 1/3).
(ii) Ratio 1:1
Let the point dividing the line internally in the ratio 1:1 be Q.
Using the section formula, the coordinates of Q(x,y) can be calculated as:
x = (1 * x2 + 1 * x1) / (1 + 1)
y = (1 * y2 + 1 * y1) / (1 + 1)
Substituting the coordinates of A(1,3) and B(-4,-1), we find:
x = (1 * (-4) + 1 * 1) / 2
y = (1 * (-1) + 1 * 3) / 2
Simplifying, we get:
x = ( -4 + 1) / 2 = -3/2
y = ( -1 + 3) / 2 = 1
So, the point which divides the line joining A and B internally in the ratio 1:1 is Q(-3/2, 1).
(iii) Ratio 3:1
Let the point dividing the line internally in the ratio 3:1 be R.
Using the section formula, the coordinates of R(x,y) can be calculated as:
x = (1 * x2 + 3 * x1) / (3 + 1)
y = (1 * y2 + 3 * y1) / (3 + 1)
Substituting the coordinates of A(1,3) and B(-4,-1), we find:
x = (1 * (-4) + 3 * 1) / 4
y = (1 * (-1) + 3 * 3) / 4
Simplifying, we get:
x = ( -4 + 3) / 4 = -1/4
y = ( -1 + 9) / 4 = 2
So, the point which divides the line joining A and B internally in the ratio 3:1 is R(-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 follow these steps on a coordinate grid:
1. Draw the x-y axes.
2. Mark the point A(1,3) on the grid.
3. Mark the point B(-4,-1) on the grid.
4. Join the points A and B with a straight line. This line represents the line joining points A and B.
Now, to plot the points P(-7/3, 1/3), Q(-3/2, 1), and R(-1/4, 2):
5. Mark the point P(-7/3, 1/3) on the grid.
6. Mark the point Q(-3/2, 1) on the grid.
7. Mark the point R(-1/4, 2) on the grid.
You should now have the line AB and the points P, Q, and R plotted on the graph.