Question

A and B are two points on the $$x$$-axis and $$y$$-axis respectively.

1. Write down the coordinates of A and B. 
2. P is a point on AB such that $$\mathrm{AP} : \mathrm{PB} = 3 : 1$$. Using section formula, find the coordinates of point P. 
3. Find the equation of a line passing through P and perpendicular to AB.

Answer 1

From the standard figure: Let the coordinates be $$\mathrm{A}(4, 0)$$ and $$\mathrm{B}(0, 4)$$.

i. The coordinates of the points are:

$$\mathrm{A} = (4, 0)$$

$$\mathrm{B} = (0, 4)$$

ii. $$\mathrm{P}$$ divides $$\mathrm{AB}$$ in the ratio $$3 : 1$$:

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

$$= \frac{4}{4}$$

$$= 1$$

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

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

$$= 3$$ 

So, $$\mathrm{P} = (1, 3)$$.

iii. Slope of $$\mathrm{AB}$$:

$$m_{\mathrm{AB}} = \frac{4 - 0}{0 - 4}$$

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

$$= -1$$

Slope of the perpendicular line:

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

Equation of the line passing through $$\mathrm{P}(1, 3)$$: 

$$y - 3 = 1(x - 1)$$

$$y - 3 = x - 1$$

$$x - y + 2 = 0$$