Question

If the lines 2x + y – 3 = 0, 5x + ky – 3 = 0 and 3x – y – 2 = 0 are concurrent, find the value of k.
(a) 1
(b) – 1
(c) 2
(d) – 2

Answer 1

Hint: First look at the definition of concurrent. Now apply that definition to those 3 given lines. Find the condition of k. Now, try to bring all the terms with k on to the left-hand side and all constants to the right-hand side. At this position, find the coefficient of k and divide with it on both sides. Then, we will get the value of k which is the required result.

Complete step-by-step answer:
Concurrent: When a group of lines passes through a single point, they are said to be concurrent and their intersection is called a point of concurrency.
Given 3 lines in the question that is written as:
\[2x+y-3=0.....\left( i \right)\]
\[5x+ky-3=0.....\left( ii \right)\]
\[3x-y-2=0.....\left( iii \right)\]
By definition of concurrent lines, we can say that the line equation (ii) passes through the intersection of equation (i) and (iii).
Finding the intersection of equation (i) and (iii) is done by the substitution method. By adding y on both the sides of the equation (iii), we get it as:
\[y=3x-2....\left( iv \right)\]
By substituting equation (iv) into equation (i), we get the equation as:
\[\Rightarrow 2x+\left( 3x-2 \right)-3=0\]
By taking x common from the first two terms, we get it as:
\[\Rightarrow 5x-5=0\]
By adding 5 on both the sides, we get the equation as:
\[\Rightarrow 5x=5\]
By dividing with 5 on both the sides, we get the value of x as:
\[\Rightarrow x=1\]
By substituting this into equation (iv), we get it as follows:
\[y=3\left( 1 \right)-2=1\]
From the above two equations, we can say the intersection point is (x, y) = (1, 1).
By definition, we say equation (ii) passes through (1, 1). By substituting this equation (ii), we get it as
\[\Rightarrow 5\left( 1 \right)+k\left( 1 \right)-3=0\]
By simplifying the equation, we can write it as:
\[\Rightarrow k+2=0\]
By subtracting 2 on both the sides, we get the equation as:
\[\Rightarrow k+2-2=-2\]
By simplifying the above equation, we get it in the form of:
\[\Rightarrow k=-2\]
The value of k for which the given lines are concurrent is – 2.
Hence, option (d) is the right answer

Note: The intersection can also be found by the elimination method by just adding both the equations and y terms can be eliminated thus getting the value of x from which you can find y value. Anyways you get the same intersection point thus getting the same value for k. Be careful while substituting as it is an important step.