Three lines are given in parametric form as follows:
Line 1: \(x = x_{1}+a_{1}t_1, \; y = y_{1}+b_{1}t_1\)
Line 2: \(x = x_{2}+a_{2}t_2, \; y = y_{2}+b_{2}t_2\)
Line 3: \(x = x_{3}+a_{3}t_3, \; y = y_{3}+b_{3}t_3\)
They are said to be concurrent if there exists a unique point at which all three lines intersect.
(a) Solve the system of equations formed by equating the x and y components of these parametric equations to find the coordinates of the point of concurrency in terms of the parameters, assuming the lines are concurrent.
(b) Explain why the solution obtained in part (a) corresponds to a single unique intersection point for concurrent lines. Describe the conditions on the system of equations that ensure this uniqueness.
Question
Solve the system of equations to find the point of concurrency by equating the x and y components of the three parametric equations. Express the intersection point in terms of the parameters.
Hint
Hint
Hint
Form pairs of equations by equating x-components and y-components from different lines. Solve this system to find the values of the parameters \(t_1, t_2, t_3\).
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Complete Answer
The lines intersect at the point \( (h,k) \) where \( h = x_{1}+a_{1}t_1 \) and \( k = y_{1}+b_{1}t_1 \), and \( t_1, t_2, t_3 \) are the solutions to the system of equations:
\(x_{1}+a_{1}t_1 = x_{2}+a_{2}t_2\)
\(x_{1}+a_{1}t_1 = x_{3}+a_{3}t_3\)
\(y_{1}+b_{1}t_1 = y_{2}+b_{2}t_2\)
\(y_{1}+b_{1}t_1 = y_{3}+b_{3}t_3\)
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Question
Justify why the solution from part (a) represents a single unique intersection point for concurrent lines. What conditions on the system of equations guarantee this uniqueness?
Hint
Hint
Hint
Consider the number of solutions to the system of linear equations. Relate a unique solution to the geometric condition of concurrency. What must be true about the system for there to be exactly one solution?
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Answer
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Explanation
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Complete Answer
For concurrent lines, the system of equations derived from equating parametric forms must have a unique solution for the parameters \( t_1, t_2, t_3 \). This unique solution ensures that there is exactly one set of parameter values that satisfy all line equations simultaneously, corresponding to a single point \( (h,k) \) where all three lines intersect. The condition for uniqueness is that the system is consistent and independent, meaning there is at least one solution and the solution is unique.
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