Question

The two points (2, 1) and (3, -1) with respect to line 3x - 5y + 7 = 0 is:
1. On the line
2. On the same side of the line
3. On opposite side of the line
4. None

Answer 1

Hint: To find the position of any two points with respect to any line, put the coordinate value (i.e. points) in the line given and check the sign of the magnitude. If value obtained by putting both the points in line are 0, then we can say that points lies on the line itself, but if the values are not zero and both are of same sign, then we can say that point lies on same side of the line, and if the values so obtained are of different sign then both the points lies on the opposite side.

Complete step-by-step answer:
The first step to find the position of the two given points (2, 1) and (3, -1) with respect to given line 3x - 5y + 7 = 0, is to put the points in the given line.
Let the given line be \[P\left( x,\text{ }y \right)\text{ }=\text{ }3x\text{ }-\text{ }5y\text{ }+\text{ }7\]
Now, first put the point (2, 1) in the above equation.
So,\[P\text{ }\left( 2,\text{ }1 \right)=3\left( 2 \right)5\left( 1 \right)+7\]
\[\Rightarrow ~P\left( 2,\text{ }1 \right)\text{ }=\text{ }8\]
Now, by putting the point (3, -1) in the above equation, we will get:
$P\left( 3,-1 \right)=3\left( 3 \right)-5\left( -1 \right)+7$
$\Rightarrow P\left( 3,-1 \right)=9+5+7$
$\Rightarrow P\left( 3,-1 \right)=21$
Now, we also know that if value obtained by putting both the points in line are 0, then points lies on the line itself, but if the values are not zero and both are of same sign, then point lies on same side of the line, and if the values so obtained are of different sign then both the points lies on the opposite side of line .
And, here we have seen that the value obtained after putting the points in the line are not zero, and both have the same magnitude, so the points will lie on the same side of the line.
Hence, option (2) is our required answer.

Note: While putting the value of points students are required to take special care of the sign of the magnitude so obtained, otherwise they will get the wrong answer. The above question can also be solved by plotting the line and the points on the graph.

From the graph we can see that both the point (2, 1) and (3, -1) lies on the same side of the line 3x-5y+7 =0