Physics and Mathematics
Position of a Point w.r.t. to a Given Line
Practice Questions with Step-by-Step Solutions
Question 1: Determine the position of the point [(2, 1)] with respect to the line [x + y − 4 = 0].
Step-by-Step Solution:
1. Write the given line in the form:
[ax + by + c = 0]
Here, [a = 1,][ b = 1,][ c = −4]
2. Substitute point [(x_1, y_1) = (2, 1)] into [ax + by + c]:
[S = a·x_1 + b·y_1 + c]
[S = 1·2 + 1·1 − 4 = −1]
3. Interpret the result:
* If [S > 0], point lies on the positive side
* If [S < 0], point lies on the negative side
* If [S = 0], point lies on the line
4. Since [S = −1 < 0], the point lies on the **negative side** of the line.
Conclusion: The point [(2, 1)] lies on the negative side of the line.
Question 2: Find the position of the point [(1, 3)] with respect to the line [2x − y − 1 = 0].
Step-by-Step Solution:
1. Identify coefficients:
[a = 2,][ b = −1,][ c = −1]
2. Substitute the point:
[S = 2·1 − 1·3 − 1 = −2]
3. Since [S < 0], the point lies on the negative side of the line.
Conclusion: The point [(1, 3)] lies on the negative side.
Question 3: Determine the position of [(3, 2)] with respect to the line [3x + 2y − 13 = 0].
Step-by-Step Solution:
1. Coefficients:
[a = 3,][ b = 2,][ c = −13]
2. Compute:
[S = 3·3 + 2·2 − 13 ][= 13 − 13 = 0]
3. Since [S = 0], the point lies exactly on the line.
Conclusion: The point [(3, 2)] lies on the given line.
Question 4: Find the position of [(−1, 2)] with respect to the line [x − 2y + 1 = 0].
Step-by-Step Solution:
1. Coefficients:
[a = 1,][ b = −2,][ c = 1]
2. Substitute point:
[S = 1·(−1) − 2·2 + 1 ][= −4]
3. Since [S < 0], the point lies on the negative side.
Conclusion: The point [(−1, 2)] lies on the negative side.
Question 5: Determine the position of [(4, −1)] with respect to the line [2x + y − 5 = 0].
Step-by-Step Solution:
1. Coefficients:
[a = 2,][ b = 1,][ c = −5]
2. Substitute:
[S = 2·4 + 1·(−1) − 5 ][= 2]
3. Since [S > 0], the point lies on the positive side.
Conclusion: The point [(4, −1)] lies on the positive side.
Question 6: Find the position of the origin with respect to the line [3x − 4y + 6 = 0].
Step-by-Step Solution:
1. Origin coordinates: [(0, 0)]
2. Substitute:
[S = 3·0 − 4·0 + 6 ][= 6]
3. Since [S > 0], the origin lies on the positive side.
Conclusion: The origin lies on the positive side of the line.
Question 7: Determine the position of [(1, 1)] with respect to the line [y = x].
Step-by-Step Solution:
1. Write line in standard form:
[x − y = 0]
2. Substitute point:
[S = 1 − 1 = 0]
3. Since [S = 0], the point lies on the line.
Conclusion: The point [(1, 1)] lies on the line.
Question 8: Find the position of [(−2, −3)] with respect to the line [x + y + 1 = 0].
Step-by-Step Solution:
1. Substitute:
[S = −2 − 3 + 1 ][= −4]
2. Since [S < 0], the point lies on the negative side.
Conclusion: The point lies on the negative side.
Question 9: Find the position of [(a, b)] with respect to the line [lx + my + n = 0].
Step-by-Step Solution:
1. Substitute point:
[S = la + mb + n]
2. Interpretation:
* If [la + mb + n > 0] → positive side
* If [la + mb + n < 0] → negative side
* If [la + mb + n = 0] → point lies on the line
Conclusion: Position depends on the sign of [la + mb + n].
Question 10: If a point [(2, 3)] lies on the positive side of the line [ax + by − 7 = 0], find the condition on [a] and [b].
Step-by-Step Solution:
1. Substitute the point:
[S = 2a + 3b − 7]
2. For positive side:
[2a + 3b − 7 > 0]
Conclusion: Required condition is
[2a + 3b > 7].
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