Question

A straight line passes through the points P(–1, 4) and Q(5, –2). It intersects $$x$$-axis and $$y$$-axis at the points A and B respectively and M is the mid-point of AB.
Find: 

1. the equation of the line;
2. the co-ordinates of A and B;
3. the co-ordinates of M.

Answer 1

i. Slope of line $$PQ$$: 

$$m = \frac{-2 - 4}{5 - (-1)}$$

$$= \frac{-6}{6}$$

$$= -1$$

Equation of the line passing through $$P(-1, 4)$$: 

$$y - 4 = -1(x - (-1))$$

$$y - 4 = -(x + 1)$$

$$y - 4 = -x - 1$$

$$x + y - 3 = 0$$

ii. For intersection with $$x$$-axis (point $$A$$), set $$y = 0$$: 

$$x + 0 - 3 = 0$$

$$\implies x = 3$$ 

So, the coordinates of $$A$$ are $$(3, 0)$$.

For intersection with $$y$$-axis (point $$B$$), set $$x = 0$$: 

$$0 + y - 3 = 0$$

$$\implies y = 3$$

So, the coordinates of $$B$$ are $$(0, 3)$$.

iii. Since $$M$$ is the mid-point of $$AB$$: 

$$M = \left(\frac{3 + 0}{2}, \frac{0 + 3}{2}\right) = \left(\frac{3}{2}, \frac{3}{2}\right)$$ or $$(1.5, 1.5)$$