Question

The line $2x + y - 3 = 0$ divides the line segment joining the points $A(1,2)$ and $B(2,-1)$ in the ratio $a:b$ at the point $C$. If the point $C$ divides the line segment joining the points $P\left( \frac{b}{3a}, -3 \right)$ and $Q\left( -3, \frac{-b}{3a} \right)$ in the ratio $p:q$, then $\frac{p}{q} + \frac{q}{p} =$:

• A. $\frac{29}{10}$
• B. $\frac{17}{10}$
• C. 6
• D. 5

Hint:

Apply the section rule and the corresponding line division formulas to tackle problems involving the division of line segments.

Answer 1

The Correct Option is A

Consider the segment joining the points $A(1,2)$ and $B(2,-1)$. The equation of the line that divides this segment in the ratio $a:b$ is given by: $$\left( \frac{b}{3a}, -3 \right) \quad \text{and} \quad \left( -3, \frac{-b}{3a} \right).$$ Since the line divides the segment in the ratio $p:q$, this relationship holds.