Problem Statement
The line segment joining $P(3, 3)$ and $Q(6, -6)$ is intersected by $x$-axis at $A$. Find:
(i) Ratio in which $A$ divides $PQ$.
(ii) Coordinates of $A$.
(iii) Equation of line through $A$ perpendicular to $PQ$.
Verified Solution & Marking Scheme
Find Ratio
$y_A = \frac{k(-6) + 3}{k + 1} = 0 \implies k = 1/2$. Ratio is $1 : 2$.
Find Coordinates
$x_A = \frac{1(6) + 2(3)}{3} = 4 \implies A(4, 0)$.
Perpendicular Line
Slope of $PQ = \frac{-9}{3} = -3 \implies m_\perp = \frac{1}{3}$. Line: $y - 0 = \frac{1}{3}(x - 4) \implies x - 3y - 4 = 0$.