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

Hint: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 \[{180^0}\].