Question

Choose the correct statement which describes the position of the point $\left( { - 6,2} \right)$ relative to straight lines $2x + 3y - 4 = 0$ and $6x + 9y + 8 = 0$.
A. Below both the lines
B. Above both the lines
C. In between the lines
D. None of the above.

Answer 1

The given question is based on the topic “straight lines”. A straight line is every point on the line segment joining any two points on it. So, every first degree equation in $x$ and $y$ represents a straight line. Here a point is given and asked us to find the position of that point to the given straight lines. So let's substitute the points in the given straight lines to get a value. Now if the final result is above $0$, then the point is above both the straight lines, if the final result is below $0$, then the point is below both the straight lines and if the final result is equal to $0$, then the point is in between the straight lines.

Complete step by step answer:
The given point is $\left( { - 6,2} \right)$, we know that the points in a graph is of the form $(x,y)$, where $x$ represent $x$-axis and $y$represents $y$-axis. Thus the point $\left( { - 6,2} \right)$ means that $x = - 6,y = 2$.
The given straight lines is:
Line 1: ${L_1} = 2x + 3y - 4 = 0$
Line 2: ${L_2} = 6x + 9y + 8 = 0$.
Now substitute the point $\left( { - 6,2} \right)$ in both straight lines, ${L_1}$ and ${L_2}$.
Substituting $\left( { - 6,2} \right)$ in ${L_1}$
${L_1}( - 6,2) \Rightarrow 2( - 6) + 3(2) - 4 = 0$
$- 12 + 6 - 4$
By subtracting $4$from $6$ we will get 2$$$$(6 - 4 = 2).
$- 12 + 2$

Now, there are two numbers of different signs. We know that $- \times + = -$.Therefore, perform subtraction and put the greatest number sign to the answer. Here $12$ is the largest number and its sign is $-$ $( - 12 + 2 = - 10)$.
$- 10$
${L_1}( - 6,2) \Rightarrow 2( - 6) + 3(2) - 4 = - 10 < 0$.
Now substituting $\left( { - 6,2} \right)$ in ${L_2}$
${L_2}( - 6,2) \Rightarrow 6( - 6) + 9(2) + 8 = 0$
$- 36 + 18 + 8$
First let’s add $18$ and $8$ we will get $26$ $(18 + 8 = 26)$.
$- 36 + 26$
We know that $- \times + = -$.Therefore, perform subtraction and put the greatest number sign to the answer. Here $36$ is the largest number and its sign is $-$ $( - 36 + 26 = - 10)$.
${L_2}( - 6,2) \Rightarrow 6( - 6) + 9(2) + 8 = - 10 < 0$.
We can observe that the point $\left( { - 6,2} \right)$ is less than $0$, that is $- 10$ for both the lines ${L_1}$ and ${L_2}$. Thus we can say that the point $\left( { - 6,2} \right)$ lies below both the lines ${L_1}$ and ${L_2}$.

Hence, the option A is correct.

Note: Remember that the straight line is not a curve and it takes the form $ax + by + c = 0$, where $a,b{\text{ and }}c$ are the constants and $a$ and $b$ is the coefficients of $x$-axis and $y$-axis of the graph respectively. For example: $x + y = 0$, $3x - y + 2 = 0$, $y - 1 = 0$, $y = 0$. These examples show the straight lines of different values. The angle of a straight line is ${180^0}$.