Question

Calculate the ratio in which the line segment joining \(\mathrm{A}(-4, 2)\) and \(\mathrm{B}(3, 6)\) is divided by the point \(\mathrm{P}(x, 3)\). Also, find (i) x (ii) length of AP.

Answer 1

Let $$P$$ divide AB in the ratio $$m : n$$.

Using the y-coordinate of P which is 3:

$$\frac{m \times 6 + n \times 2}{m + n} = 3$$

$$\Rightarrow 6m + 2n = 3m + 3n$$

$$\Rightarrow 3m = n$$

$$\Rightarrow m : n = 1 : 3$$

Hence $$AP : PB = 1 : 3$$.

(i) Using the x-coordinate:

$$x = \frac{1 \times 3 + 3 \times (-4)}{1 + 3}$$

$$= \frac{3 - 12}{4}$$

$$= -\frac{9}{4}$$

(ii) $$A(-4, 2)$$ and $$P\left(-\frac{9}{4}, 3\right)$$:

$$AP = \sqrt{\left(-\frac{9}{4} + 4\right)^2 + (3 - 2)^2}$$

$$= \sqrt{\left(\frac{7}{4}\right)^2 + 1}$$

$$= \sqrt{\frac{49}{16} + \frac{16}{16}}$$

$$= \sqrt{\frac{65}{16}}$$

$$= \frac{\sqrt{65}}{4}\text{ units}$$