Correct Answer - Option 3 : x - y = 0
Concept:
Equation of family of lines passing through the intersection of two lines S1 = 0 and S2 = 0 is given by S1 + λS2 = 0
Calculations:
Given, the equation of lines are \(\dfrac{x}{a}+\dfrac{y}{b}=1\) and \(\dfrac{x}{b} + \dfrac{y}{a}=1\)
The equation to the straight line joining the origin to the point of intersection of the lines \(\dfrac{x}{a}+\dfrac{y}{b}=1\) and \(\dfrac{x}{b} + \dfrac{y}{a}=1 \) is
\((\dfrac{x}{a}+\dfrac{y}{b}-1) + λ (\dfrac{x}{b} + \dfrac{y}{a}-1) = 0\) ....(1)
which is passing through the origin.
⇒ \((\dfrac{0}{a}+\dfrac{0}{b}-1) - λ (\dfrac{0}{b} + \dfrac{0}{a}-1) = 0\)
⇒ \(λ = -1\)
Equation of line becomes,
⇒ \((\dfrac{x}{a}+\dfrac{y}{b}-1) +(-1) (\dfrac{x}{b} + \dfrac{y}{a}-1) = 0\)
⇒ \((\dfrac{x}{a}+\dfrac{y}{b}-1)-(\dfrac{x}{b} + \dfrac{y}{a}-1) = 0\)
⇒ \(\dfrac{x}{a}+\dfrac{y}{b}-1 - \dfrac{x}{b} - \dfrac{y}{a}+1 = 0\)
⇒ \(\rm x(\dfrac{1}{a}-\dfrac{1}{b})- y(\dfrac{1}{a} - \dfrac{1}{b}) = 0\)
⇒ x - y = 0
Hence, the equation to the straight line joining the origin to the point of intersection of the lines \(\dfrac{x}{a}+\dfrac{y}{b}=1\) and \(\dfrac{x}{b} + \dfrac{y}{a}=1 \) is x - y = 0.