Question

The point which does not lie in the half plane $2x + 3y - 12 \leqslant 0$ is
A: $(1,2)$
B: $(2,1)$
C: $(2,3)$
D: $( - 3,2)$

Answer 1

Here, they have given the half plane equation as $2x + 3y - 12 \leqslant 0$, for this we have to substitute the values of $x$ and $y$ which have been given in the options. If the equation satisfies the values then that point will be in half plane, if the points does not satisfy the condition they have given then that point does not lie in the half plane.

Complete step by step solution:
They have given the half plane equation as $2x + 3y - 12 \leqslant 0$, now we need to check out of the given points which point does not lie in the half plane. So, for that we need to substitute each and every value of $x$ and $y$ in the given equation.
Let us take first option that is $x = 1$ and $y = 2$, substitute these values in given half plane equation, we get
$(2 \times 1) + (3 \times 2) - 12 \leqslant 0$
$\Rightarrow 2 + 6 - 12 \leqslant 0$
$\Rightarrow 8 - 12 \leqslant 0$
$\Rightarrow - 4 \leqslant 0$
Hence, $- 4$ is less than $0$, it satisfies the given condition, so it lies in half-plane.
Let us take second option that is $x = 2$ and $y = 1$, substitute these values in given half plane equation, we get
$(2 \times 2) + (3 \times 1) - 12 \leqslant 0$
$\Rightarrow 4 + 3 - 12 \leqslant 0$
$\Rightarrow 7 - 12 \leqslant 0$
$\Rightarrow - 3 \leqslant 0$
Therefore, $- 3$ is less than $0$.
Hence it satisfies the given condition, so it lies in half-plane.
Let us take option C that is $x = 2$ and $y = 3$, substitute these values in given half plane equation, we get
$(2 \times 2) + (3 \times 3) - 12 \leqslant 0$
$\Rightarrow 4 + 9 - 12 \leqslant 0$
$\Rightarrow 13 - 12 \leqslant 0$
$\Rightarrow 1 \leqslant 0$
Therefore, $1$ is greater than $0$.
Hence it does not satisfy the given condition, so it does not lie in half-plane.
Option C is the correct answer.
We will also check for option D if it lies in the half plane equation or not.
Let us take option D that is $x = - 3$ and $y = 2$, substitute these values in given half plane equation, we get
$(2 \times - 3) + (3 \times 2) - 12 \leqslant 0$
$\Rightarrow - 6 + 6 - 12 \leqslant 0$
$\Rightarrow 6 - 18 \leqslant 0$
$\Rightarrow - 12 \leqslant 0$
Therefore, $- 12$ is less than $0$.
Hence it satisfies the given condition, so it lies in half-plane.

There, we can conclude that the option C $(2,3)$ is the correct answer, which does not lie in $2x + 3y - 12 \leqslant 0$.

Note:
Whenever they have given an equation to check for, there may be two options which are correct. So, if any one option is satisfied at first itself, we should not stop solving it itself. We should check for all the options so that we will end up finding the correct answer.