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.