Burns Summers
05/19/2024 · Primary School

16. a) Two equation of straight lines are given by \( L_{1}: a_{1} x+b_{1} y+c_{1}=0 \) \( L_{2}: a_{2} x+b_{2} y+c_{2}=0 \) i) Write the condition that the above lines \( L_{1} \) and \( L_{2} \) are perpendicular. iii) Write the expression for the perpendicular distance from point \( \left(x_{1}, y_{1}\right) \) to line \( L_{1} \)

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**i) Condition for Perpendicular Lines** Two lines are perpendicular if the product of their slopes is \(-1\). For the given lines \( L_{1} \) and \( L_{2} \), this means: \[ a_{1}a_{2} + b_{1}b_{2} = 0 \] **iii) Perpendicular Distance from Point to Line** The distance from point \( (x_{1}, y_{1}) \) to line \( L_{1} \) is: \[ D = \frac{|a_{1}x_{1} + b_{1}y_{1} + c_{1}|}{\sqrt{a_{1}^{2} + b_{1}^{2}}} \]

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