Question

In what ratio does the point \(\mathrm{P}(a, -1)\) divide the join of \(\mathrm{A}(1, -3)\) and \(\mathrm{B}(6, 2)\). Hence, find the value of a.

Answer 1

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

Using the y-coordinate of P which is $$-1$$:

$$\frac{k \times 2 + 1 \times (-3)}{k + 1} = -1$$

$$\Rightarrow 2k - 3 = -k - 1$$

$$\Rightarrow 3k = 2$$

$$\Rightarrow k = \frac{2}{3}$$

Hence the required ratio is $$\dfrac{2}{3} : 1 = 2 : 3$$.

Now the x-coordinate gives the value of $$a$$:

$$a = \frac{2 \times 6 + 3 \times 1}{2 + 3}$$

$$= \frac{12 + 3}{5}$$

$$= \frac{15}{5}$$

$$= 3$$

Hence $$a = 3$$.