Question

A line segment AB meets $$x$$-axis at A and $$y$$-axis at B. P(4, –1) divides AB in the ratio 1 : 2.

1. Find the co-ordinates of A and B. 
2. Find the equation of the line through P and perpendicular to AB.

Answer 1

i. Let the coordinates of $$\mathrm{A}$$ on the $$x$$-axis be $$(a, 0)$$ and $$\mathrm{B}$$ on the $$y$$-axis be $$(0, b)$$. 

Given that $$\mathrm{P}(4, -1)$$ divides $$\mathrm{AB}$$ in the ratio $$1 : 2$$: 

$$4 = \frac{1(0) + 2(a)}{1 + 2} = \frac{2a}{3}$$

$$\implies 2a = 12$$

$$\implies a = 6$$ 

$$-1 = \frac{1(b) + 2(0)}{1 + 2} = \frac{b}{3}$$

$$\implies b = -3$$ 

So, the coordinates are $$\mathrm{A}(6, 0)$$ and $$\mathrm{B}(0, -3)$$.

ii. Slope of line $$\mathrm{AB}$$: 

$$m_{\mathrm{AB}} = \frac{-3 - 0}{0 - 6}$$

$$= \frac{-3}{-6}$$

$$= \frac{1}{2}$$

Slope of the perpendicular line through $$\mathrm{P}$$: 

$$m = -\frac{1}{m_{\mathrm{AB}}} = -2$$

Equation of the line passing through $$\mathrm{P}(4, -1)$$: 

$$y - (-1) = -2(x - 4)$$

$$y + 1 = -2x + 8$$

$$2x + y - 7 = 0$$