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If a linear equation has solutions $\left( { - 2,2} \right),\,\left( {0,0} \right),\,\left( {2, - 2} \right),$ then it is of the form :(A) $y - x = 0$(B) $x + y = 0$(C) $- 2x + y = 0$(D) $- x + by = 0$

Last updated date: 13th Jun 2024
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Hint: In this first we will assume a linear equation in two variables. After this we will put the value of points which are given in the question and then from this we will find three equations having three unknown variables. Now, with the help of these three equations we can easily find the three unknown variables by using elimination method. And then we can find the form of the equation.

Complete step-by-step solution:
Let’s assume the linear equation is of the form $ax + by + c = 0$ .Now, we will put the value of the solutions given in the question i.e. $\left( { - 2,2} \right),\,\left( {0,0} \right),\,\left( {2, - 2} \right)$ .
First of all put $\left( { - 2,2} \right)$ i.e. $x = - 2$ and $y = 2$ in the equation $ax + by + c = 0$. Therefore, we will get:
$ax + by + c = 0 \\ \Rightarrow - 2a + 2b + c = 0\_\_\_\left( 1 \right)$
Now, put $\left( {0,0} \right)$ i.e. $x = 0$ and $y = 0$ in the equation $ax + by + c = 0$. Therefore, we will get:
$ax + by + c = 0 \\ \Rightarrow c = 0\_\_\_\left( 2 \right)$
Now, put $\left( {2, - 2} \right)$ i.e. $x = 2$ and $y = - 2$ in the equation $ax + by + c = 0$. Therefore, we will get:
$ax + by + c = 0 \\ \Rightarrow 2a - 2b + c = 0\_\_\_\left( 3 \right)$
Now, put the value of $c = 0$ in the equation $\left( 3 \right)$ .
$\Rightarrow 2a - 2b + 0 = 0 \\ \Rightarrow 2a = 2b$
Therefore, we got:
$\Rightarrow a = b$
Now, put the value of $a = b$ and $c = 0$ in the equation $ax + by + c = 0$ . Therefore, we will get:
$ax + by + c = 0 \\ \Rightarrow ax + ay + 0 = 0$
Now, after simplifying the above equation we will get
$\Rightarrow a\left( {x + y} \right) = 0 \\ \Rightarrow x + y = 0$
Therefore, we found that the equation $x + y = 0$ has solutions $\left( { - 2,2} \right),\,\left( {0,0} \right),\,\left( {2, - 2} \right)$ .

Hence, the correct option is (B).

Note: The important thing in this question is the assumption of the linear equation in two variables. Because the solutions of the equation are in two variables. We should have the idea that in order to find three unknowns we need three equations. So be careful about these things while solving these types of questions.