a) Find the ratio in which the point P (-1, a) divides the joining of points A(-5, 4) and B (3, -2). Hence, find a.

b)
The line joining the points A(4, -5) and B (4, 5) is divided by the point P, such that AP: PB = 2 : 3. Find the coordinates of point P.

[6 MARKS]

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Solution

Part a) : 4 Marks
Part b): 2 Marks

a) Let the required ratio be k:1

Using section formula, we have
$- 1 = \frac{1 \times (- 5) + k \times 3}{k + 1} \Rightarrow - k - 1 = - 5 + 3 k \Rightarrow - 4 k = - 4$
So, k=1 and ratio = 1:1
Now, $a = \frac{1 \times 4 + 1 \times (- 2)}{2} \Rightarrow a = 1$

b)
Using section formula, we have
$x = \frac{m_{1} x_{2} + m_{2} x_{1}}{m_{1} + m_{2}} = \frac{2 \times 4 + 3 \times 4}{2 + 3} = \frac{8 + 12}{5} = 4 y = \frac{m_{1} y_{2} + m_{2} y_{1}}{m_{1} + m_{2}} = \frac{2 \times 5 + 3 \times (- 5)}{2 + 3} = \frac{10 - 15}{5} = - 1 S o, P (x, y) = (4, - 1)$

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