If the diagonals of the quadrilateral formed by the lines l₁x + m₁y + n₁ = 0, l₂x + m₂y + n₂ = 0, l₁x + m₁y + n₁' = 0 and l₂x + m₂y + n₂' = 0 are perpendicular, then write the value of l₁² − l₂² + m₁² − m₂².

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Solution

The given lines are

l₁x + m₁y + n₁ = 0 ... (1)

l₂x + m₂y + n₂ = 0 ... (2)

l₁x + m₁y + n₁' = 0 ... (3)

l₂x + m₂y + n₂' = 0 ... (4)

Let (1), (2), (3) and (4) represent the sides AB, BC, CD and DA, respectively.

The equation of diagonal AC passing through the intersection of (2) and (3) is given by
l₁x + m₁y + n₁' + $λ$ (l₂x + m₂y + n₂) = 0

$\Rightarrow (l_{1} + λ l_{2}) x + (m_{1} + λ m_{2}) y + (n_{1}' + λ n_{2}) = 0 \Rightarrow Slope of diagonal A C = - (\frac{l_{1} + λ l_{2}}{m_{1} + λ m_{2}})$

Also, the equation of diagonal BD, passing through the intersection of (1) and (2), is given by
l₁x + m₁y + n₁ + $μ$ (l₂x + m₂y + n₂) = 0

$\Rightarrow (l_{1} + μ l_{2}) x + (m_{1} + μ m_{2}) y + (n_{1} + μ n_{2}) = 0 \Rightarrow Slope of diagonal B D = - (\frac{l_{1} + μ l_{2}}{m_{1} + μ m_{2}})$

The diagonals are perpendicular to each other.

∴ $(\frac{l_{1} + λ l_{2}}{m_{1} + λ m_{2}}) (\frac{l_{1} + μ l_{2}}{m_{1} + μ m_{2}}) = - 1$

$\Rightarrow (l_{1} + λ l_{2}) (l_{1} + μ l_{2}) = - (m_{1} + λ m_{2}) (m_{1} + μ m_{2}) Let λ = - 1, μ = 1 \Rightarrow (l_{1} - l_{2}) (l_{1} + l_{2}) = - (m_{1} - m_{2}) (m_{1} + m_{2}) \Rightarrow ({l_{1}}^{2} - {l_{2}}^{2}) = - ({m_{1}}^{2} - {m_{2}}^{2}) \Rightarrow ({l_{1}}^{2} - {l_{2}}^{2}) + ({m_{1}}^{2} - {m_{2}}^{2}) = 0$

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If the diagonals of the quadrilateral formed by the lines $l - 1 x + m_{1} y + n_{1} = 0, l_{2} x + m_{2} y + n_{2} = 0, l_{1} x + m_{1} y + n_{1}^{'} = 0$ and $(l_{2} x + m_{2} y + n_{2}^{'} = 0$ are perpendicular,then write the value of $l_{1}^{2} - l_{2}^{2} + m_{2} - m_{2}^{2}$