Question

Find the position of the points \[\left( {3,4} \right)\] and \[\left( {2, - 6} \right)\] with respect to the line \[3x - 4y = 8\]?
A. On the same side of the line
B. On different side of the line
C. One point on the line and the other outside of the line
D. Both points on the line

Answer 1

Hint: Substitute the values of the coordinates of the points in the given equation of a line. If the signs of output are the same, then the points lie on the same side. If the signs of output are different, then the points lie on the same different sides.

Formula used:
The points \[\left( {{x_1},{y_1}} \right)\] and \[\left( {{x_2},{y_2}} \right)\] lies on the same side of a line \[ax + by + c = 0\], if \[a{x_1} + b{y_1} + c = 0\] and \[a{x_2} + b{y_2} + c = 0\] have the same signs.
The points \[\left( {{x_1},{y_1}} \right)\] and \[\left( {{x_2},{y_2}} \right)\] lies on the opposite sides of a line \[ax + by + c = 0\], if \[a{x_1} + b{y_1} + c = 0\] and \[a{x_2} + b{y_2} + c = 0\] have the opposite signs.

Complete step by step solution:
The given equation of line is \[3x - 4y = 8\] and the points are \[\left( {3,4} \right)\] and \[\left( {2, - 6} \right)\].
Substitute \[\left( {3,4} \right)\] in the equation of line.
\[3\left( 3 \right) - 4\left( 4 \right) - 8 = 0\]
\[ \Rightarrow \]\[9 - 16 - 8 = 0\]
\[ \Rightarrow \]\[ - 15 < 0\]

Substitute \[\left( {2, - 6} \right)\] in the equation of line.
\[3\left( 2 \right) - 4\left( { - 6} \right) - 8 = 0\]
\[ \Rightarrow \]\[6 + 20 - 8 = 0\]
\[ \Rightarrow \]\[18 > 0\]
Since the signs of the output are opposite. Hence, the points lie on the opposite sides of a line.
Hence the correct option is B.

Note: If any point lies on a line, then the point satisfies the equation of a line.
If a point lies outside of a line, then the point satisfies the inequality equation of a line.