Problem 1 :
Show that the following straight lines are concurrent by solving equations. And also, find the point of concurrency.
2x - 3y + 4 = 0
9x + 5y = 19
2x -7y + 12 = 0
Problem 2 :
Use determinant to show that the following straight lines are concurrent. And also, find the point of concurrency.
3x + 4y = 13
2x - 7y = -1
5x - y = 14
1. Answer :
2x - 3y + 4 = 0 ----(1)
9x + 5y - 19 = 0 ----(2)
2x -7y + 12 = 0 ----(3)
5(1) - 3(2) :
37x = 37
x = 1
Substitute x = 1 into (1).
2(1) - 3y + 4 = 0
2 - 3y + 4 = 0
-3y + 6 = 0
-3y = -6
y = 2
The point of interesection of (1) and (2) is (1, 2).
Substitute the point (1, 2) into (3).
2(1) - 7(2) + 12 = 0
2 -14 +12 = 0
-12 + 12 = 0
0 = 0 (true)
The point (1, 2) satisfies (3). So, the point (1, 2) lies on the third line.
Therefore, the given three lines are concurrent and the point of concurrency is (1, 2).
2. Answer :
Write the given equations in general form.
3x + 4y - 13 = 0 ----(1)
2x - 7y + 1 = 0 ----(2)
5x - y - 14 = 0 ----(3)
Find the value of the determinant with the coefficients of x, y and constant terms as explained in Method 2.
= 3(98 + 1) - 4(-28 - 5) - 13(-2 + 35)
= 3(99) - 4(-33) - 13(33)
= 297 + 132 - 429
= 429 - 429
= 0
Since the value of the determinant is zero, the given lines are concurrent.
Solve any two of (1), (2) and (3) to find the point of concurrency.
2(1) - 3(2) :
29y = 29
y = 1
Substitute y = 1 into (1).
3x + 4(1) - 13 = 0
3x + 4 - 13 = 0
3x - 9 = 0
3x = 9
x = 3
The point of concurrency is (3, 1).
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