Question

A straight line passes through the points A(2, – 4) and B(5, – 2).
Find:

1. the slope of the line AB;
2. the equation of the line AB;
3. the value of $$k$$, if AB passes through the point $$\mathrm{P}(k + 3, k - 4)$$.

Answer 1

i. Slope of line $$AB$$:

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

$$= \frac{-2 + 4}{3}$$

$$= \frac{2}{3}$$

ii. Equation of line $$AB$$ using point $$A(2, -4)$$:

$$y - (-4) = \frac{2}{3}(x - 2)$$

$$y + 4 = \frac{2}{3}(x - 2)$$

$$3(y + 4) = 2(x - 2)$$

$$3y + 12 = 2x - 4$$

$$2x - 3y - 16 = 0$$

iii. Since the line passes through $$\mathrm{P}(k + 3, k - 4)$$, these coordinates must satisfy the equation:

$$2(k + 3) - 3(k - 4) - 16 = 0$$

$$2k + 6 - 3k + 12 - 16 = 0$$

$$-k + 2 = 0$$

$$k = 2$$