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 \( \frac{p}{q} + \frac{q}{p} = \):
Show Hint
- \( \frac{29}{10} \)
- \( \frac{17}{10} \)
- 6
5
The Correct Option is A
Solution and Explanation
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.
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