Question

In what ratio does the point P(2, –5) divide the join of A(–3, 5) and B(4, –9)?

Answer 1

Let P divide AB in the ratio $$k :1$$.

Then the co-ordinates of P are $$\left(\frac{k \times 4 + 1 \times (-3)}{k + 1}, \frac{k \times (-9) + 1 \times 5}{k + 1}\right) = \left(\frac{4k - 3}{k + 1},\frac{5 - 9k}{k + 1}\right)$$

Using the x-coordinate: $$\frac{4k - 3}{k + 1} = 2$$

$$\Rightarrow 4k - 3 = 2k + 2$$

$$\Rightarrow 2k = 5$$

$$\Rightarrow k = \frac{5}{2}$$

Check with the y-coordinate: $$\frac{5 - 9k}{k + 1} = -2$$

$$\Rightarrow 5 - 9k = -2k - 2$$

$$\Rightarrow 7 = 7k$$

$$\Rightarrow k = \frac{5}{2}$$

Hence the required ratio is $$\dfrac{5}{2} : 1 = 5 : 2$$.