Three lines
L1 = x-y+6 = 0
L2 = 2x+y-3 = 0
L3 = x-2y+m = 0 are given. Which of the following can be the value of m if L1, L2 and L3 form a triangle.
Three lines
L1 = x-y+6 = 0
L2 = 2x+y-3 = 0
L3 = x-2y+m = 0 are given. Which of the following can be the value of m if L1, L2 and L3 form a triangle.
The correct options are
A 10
C 12
D 13
We can easily see that the three given lines are not parallel ( If the lines are parallel, their coefficients will be proportional). Three lines will form a triangle if they are not parallel and not concurrent.
We found that the given lines are not parallel. Now, if they are not concurrent also, we can say they form a triangle.
We will first find the condition for them to be concurrent and eliminate those values of m.[because we know how to find the value of m if they are concurrent]. We will find the point of intersection of first two lines and substitute that point in the third line to find m.
x-y+6 = 0
2x+y-3 = 0
3x+3 = 0
x = -1 and y = 5
(-1,5) lies on x-2y+m = 0
⇒ -1-10+m = 0
⇒ m = 11
Now we have to check if they are not parallel and concurrent if they will form a triangle. If two lines
L1 and L2 are not parallel, they will definitely meet at a point. Same thing is applicable for the pairs
(L1,L3) and (L2,L3). So we will get three points of intersection. These three points won't be the same
because they are not concurrent. So they form a triangle.
So the three lines will be concurrent m = 11. For all the other value of m, the lines won't be concurrent and parallel.
A
10
C
12
D
13
We can easily see that the three given lines are not parallel ( If the lines are parallel, their coefficients will be proportional). Three lines will form a triangle if they are not parallel and not concurrent.
We found that the given lines are not parallel. Now, if they are not concurrent also, we can say they form a triangle.
We will first find the condition for them to be concurrent and eliminate those values of m.[because we know how to find the value of m if they are concurrent]. We will find the point of intersection of first two lines and substitute that point in the third line to find m.
x-y+6 = 0
2x+y-3 = 0
3x+3 = 0
x = -1 and y = 5
(-1,5) lies on x-2y+m = 0
⇒ -1-10+m = 0
⇒ m = 11
Now we have to check if they are not parallel and concurrent if they will form a triangle. If two lines
L1 and L2 are not parallel, they will definitely meet at a point. Same thing is applicable for the pairs
(L1,L3) and (L2,L3). So we will get three points of intersection. These three points won't be the same
because they are not concurrent. So they form a triangle.
So the three lines will be concurrent m = 11. For all the other value of m, the lines won't be concurrent and parallel.
