# 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 these

Answer

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Hint:Use position of a point with relative to a line and check the condition to solve this problem.

Complete step-by-step answer:

The equations of two lines are \[2x+3y-4=0\] and \[6x+9y+8=0\]. It can be represented as shown below.

A point \[\left( {{x}_{1}},\text{ }{{y}_{1}} \right)\] will lie below the line \[ax+by+c=0\] if \[a{{x}_{1}}+b{{y}_{1}}+c<0\] and vice versa.

We will find the position of the point with respect to first line, i.e., \[2x+3y-4=0\], i.e., substitute \[\left( -6,2 \right)\] in the line equation, we get

\[2(-6)+3(2)-4\]

\[-12+6-4\]

\[-10<0\]

So, the given point \[\left( -6,2 \right)\] is below the line \[2x+3y-4=0\].

Now we will find the position of the point with respect to second line, i.e., \[6x+9y+8=0\], i.e., substitute\[\left( -6,2 \right)\]in the line equation, we get

\[6(-6)+9(2)+8\]

\[-36+18+8\]

\[-10<0\]

So, the given point \[\left( -6,2 \right)\]is below the line \[6x+9y+8=0\].

So, the point \[\left( -6,2 \right)\]is below both the given lines.

Hence the correct answer is option (a).

Note: The possibility of error is that instead of less than it can be considered greater than, i.e., if the \[a{{x}_{1}}+b{{y}_{1}}+c>0\]then the point is below the line. In this case we will get the wrong answer.

Complete step-by-step answer:

The equations of two lines are \[2x+3y-4=0\] and \[6x+9y+8=0\]. It can be represented as shown below.

A point \[\left( {{x}_{1}},\text{ }{{y}_{1}} \right)\] will lie below the line \[ax+by+c=0\] if \[a{{x}_{1}}+b{{y}_{1}}+c<0\] and vice versa.

We will find the position of the point with respect to first line, i.e., \[2x+3y-4=0\], i.e., substitute \[\left( -6,2 \right)\] in the line equation, we get

\[2(-6)+3(2)-4\]

\[-12+6-4\]

\[-10<0\]

So, the given point \[\left( -6,2 \right)\] is below the line \[2x+3y-4=0\].

Now we will find the position of the point with respect to second line, i.e., \[6x+9y+8=0\], i.e., substitute\[\left( -6,2 \right)\]in the line equation, we get

\[6(-6)+9(2)+8\]

\[-36+18+8\]

\[-10<0\]

So, the given point \[\left( -6,2 \right)\]is below the line \[6x+9y+8=0\].

So, the point \[\left( -6,2 \right)\]is below both the given lines.

Hence the correct answer is option (a).

Note: The possibility of error is that instead of less than it can be considered greater than, i.e., if the \[a{{x}_{1}}+b{{y}_{1}}+c>0\]then the point is below the line. In this case we will get the wrong answer.

Last updated date: 22nd Sep 2023

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