Question

(i) Points P and Q trisect the line segment joining the points A(–2, 0) and B(0, 8) such that P is near to A. Find the coordinates of P and Q.
(ii) The line segment joining the points (3, −4) and (1, 2) is trisected at the points P and Q. If the coordinates of P and Q are (p, −2) and (5/3, q) respectively. Find the values of p and q.

Answer 1

(ii) Let the points A(3, −4) and B(1, 2) is trisected at the points P(p, −2) and Q(5/3, q).

Thus, AP = PQ = QB

Therefore, P divides AB internally in the ratio 1 : 2.

Section formula: if the point (x, y) divides the line segment joining the points (x₁, y₁) and (x₂, y₂) internally in the ratio m : n, then the coordinates (x, y) = $\left(\frac{m{x}_2+n{x}_1}{m+n},\frac{m{y}_2+n{y}_1}{m+n}\right)$

Therefore, using section formula, the coordinates of P are:

$$\begin{gathered} \left(p,-2\right)=\left(\frac{1\left(1\right)+2\left(3\right)}{1+2},\frac{1\left(2\right)+2\left(-4\right)}{1+2}\right) \\ \Rightarrow \left(p,-2\right)=\left(\frac{1+6}{3},\frac{2-8}{3}\right) \\ \Rightarrow \left(p,-2\right)=\left(\frac{7}{3},\frac{-6}{3}\right) \\ \Rightarrow \left(p,-2\right)=\left(\frac{7}{3},-2\right) \\ \Rightarrow p=\frac{7}{3} \end{gathered}$$

Also, Q divides AB internally in the ratio 2 : 1.

Therefore, using section formula, the coordinates of Q are:

$$\begin{gathered} \left(\frac{5}{3},q\right)=\left(\frac{2\left(1\right)+1\left(3\right)}{2+1},\frac{2\left(2\right)+1\left(-4\right)}{2+1}\right) \\ \Rightarrow \left(\frac{5}{3},q\right)=\left(\frac{2+3}{3},\frac{4-4}{3}\right) \\ \Rightarrow \left(\frac{5}{3},q\right)=\left(\frac{5}{3},\frac{0}{3}\right) \\ \Rightarrow \left(\frac{5}{3},q\right)=\left(\frac{5}{3},0\right) \\ \Rightarrow q=0 \end{gathered}$$

Hence, the values of p and q is $\frac{7}{3}$ and 0, respectively.