Question

For the straight lines given by the equation\[\left( {2 + k} \right)x + \left( {1 + k} \right)y = 5 + 7k\] for different values of which of the following statement is true?\[k\]
A. Lines are parallel
B. Lines are passing through the point \[\left( { - 2,9} \right)\]
C. Lines are passing through the point \[\left( {2, - 9} \right)\]
D. None of these.

Answer 1

Hint: In solving the above question, first we will take all terms to one side, and simplify the terms, then we will take all non-\[k\] terms and \[k\] terms as an equation and then we will equate each term to \[0\], and then solve the two terms to get the value of \[x\], and \[y\] then we will make an ordered pair with the values to get the required point.

Complete step by step solution:
Given \[\left( {2 + k} \right)x + \left( {1 + k} \right)y = 5 + 7k\]
Now we will take all terms to one side, we will get,
\[ \Rightarrow \left( {2 + k} \right)x + \left( {1 + k} \right)y - 5 - 7k = 0\]
Now we will simplify by taking out the brackets by multiplying, we will get,
\[ \Rightarrow 2x + kx + y + ky - 5 - 7k = 0\]
Now will take all non-\[k\] terms and \[k\] terms, we will get,
\[ \Rightarrow \left( {2x + y - 5} \right) + k\left( {x + y - 7} \right) = 0\]
Now will equate each term to \[0\], we will get,
\[ \Rightarrow 2x + y - 5 = 0 - - - - (1)\]
And \[x + y - 7 = 0 - - - - (2)\]
Now subtraction equation (2) from equation (1), we get;
\[ \Rightarrow x + 2 = 0\]
\[ \Rightarrow x = - 2\]
Now substitute the value of \[x\] in equation (2), we get;
\[ \Rightarrow - 2 + y - 7 = 0\]
\[ \Rightarrow y - 9 = 0\]
\[ \Rightarrow y = 9\]
So, the values of \[x\] and \[y\] are\[\left( { - 2,9} \right)\].

The correct option is B.

Note: Another way to solve the question: Put \[\left( { - 2,9} \right)\] in the given equation \[\left( {2 + k} \right)x + \left( {1 + k} \right)y = 5 + 7k\].
\[\left( {2 + k} \right)\left( { - 2} \right) + \left( {1 + k} \right) \cdot 9 = 5 + 7k\]
\[ \Rightarrow - 4 - 2k + 9 + 9k = 5 + 7k\]
\[ \Rightarrow 5 + 7k = 5 + 7k\]
Since \[\left( { - 2,9} \right)\] satisfies the given equation, so the lines passes through the point \[\left( { - 2,9} \right)\].