Find the coordinates of the points which divide the line segment joining A(– 2, 2) and B(2, 8) into four equal parts.
Step-by-Step Solution
Key Concept: Use the section formula (internal division) to find points that divide a line segment in a given ratio. For a point P dividing AB in the ratio m:n (AP : PB = m:n), the coordinates are $$P\left(\frac{n x_1 + m x_2}{m+n},\;\frac{n y_1 + m y_2}{m+n}\right)$$ where A$(x_1,y_1)$ and B$(x_2,y_2)$. For four equal parts, the required points correspond to ratios 1:3, 2:2 (mid‑point), and 3:1.
1. Identify the end points: \(A(-2,2)\) and \(B(2,8)\).
2. Determine the ratios for the three interior points that split \(AB\) into four equal segments:
- First point \(P_1\): \(AP_1 : P_1B = 1 : 3\).
- Second point \(P_2\) (mid‑point): \(AP_2 : P_2B = 2 : 2 = 1 : 1\).
- Third point \(P_3\): \(AP_3 : P_3B = 3 : 1\).
3. Apply the section formula for each ratio.
- For \(P_1\) (ratio 1:3):
$$x_{P_1}=\frac{3\cdot(-2)+1\cdot 2}{1+3}=\frac{-6+2}{4}= -1,$$
$$y_{P_1}=\frac{3\cdot 2+1\cdot 8}{4}=\frac{6+8}{4}=\frac{14}{4}=\frac{7}{2}.$$
- For \(P_2\) (mid‑point, ratio 1:1):
$$x_{P_2}=\frac{1\cdot(-2)+1\cdot 2}{2}=0,$$
$$y_{P_2}=\frac{1\cdot 2+1\cdot 8}{2}=5.$$
- For \(P_3\) (ratio 3:1):
$$x_{P_3}=\frac{1\cdot(-2)+3\cdot 2}{4}=\frac{-2+6}{4}=1,$$
$$y_{P_3}=\frac{1\cdot 2+3\cdot 8}{4}=\frac{2+24}{4}=\frac{26}{4}=\frac{13}{2}.$$
4. List the coordinates of the three points that divide \(AB\) into four equal parts.
Thus the required points are \((-1,\;\frac{7}{2})\), \((0,5)\) and \((1,\;\frac{13}{2})\).
Correct Answer: (-1, 7/2), (0, 5), (1, 13/2)