Physics and Mathematics
Example 1 – Perpendicular Distance of a Point From a Line
Practice Questions with Step-by-Step Solution
Question 1: Find the perpendicular distance of the point [(5, 2)] from the line [x + y – 7 = 0].
Step-by-Step Solution:
1. Compare with [[Ax + By + C = 0]]:
[
A = 1,; B = 1,; C = -7
]
2. Given point [(x_1, y_1) = (5, 2)]
3. Apply distance formula:
[
d = \dfrac{|1(5) + 1(2) – 7|}{\sqrt{1^2 + 1^2}}
]
4. Simplify numerator:
[
|5 + 2 – 7| = |0| = 0
]
Answer:
[d = 0]
Conclusion:
The point lies on the line.
Question 2: Find the perpendicular distance of [(−3, 4)] from [2x + y + 6 = 0].
Step-by-Step Solution:
1. [
A = 2,; B = 1,; C = 6
]
2. Apply formula:
[
d = \dfrac{|2(-3) + 1(4) + 6|}{\sqrt{4 + 1}}
]
3. Simplify:
[
|-6 + 4 + 6| = |4| = 4
]
4. Denominator:
[
\sqrt{5}
]
Answer:
[d = \dfrac{4}{\sqrt{5}}]
Question 3: Find the perpendicular distance of [(1, -1)] from [y – 2x + 3 = 0].
Step-by-Step Solution:
1. Rewrite equation:
[
-2x + y + 3 = 0
]
2. [
A = -2,; B = 1,; C = 3
]
3. Apply formula:
[
d = \dfrac{|(-2)(1) + 1(-1) + 3|}{\sqrt{4 + 1}}
]
4. Simplify:
[
|-2 – 1 + 3| = |0| = 0
]
Answer:
[d = 0]
Conclusion:
The point lies on the given line.
Question 4: Find the distance of [(2, 3)] from the line [y = -x + 1].
Step-by-Step Solution:
1. Convert to general form:
[
x + y – 1 = 0
]
2. [
A = 1,; B = 1,; C = -1
]
3. Apply formula:
[
d = \dfrac{|1(2) + 1(3) – 1|}{\sqrt{2}}
]
4. Simplify:
[
|4| = 4
]
Answer:
[d = \dfrac{4}{\sqrt{2}}]
Question 5: Find the perpendicular distance of [(−1, −2)] from [3x – 4y + 10 = 0].
Step-by-Step Solution:
1. [
A = 3,; B = -4,; C = 10
]
2. Apply formula:
[
d = \dfrac{|3(-1) + (-4)(-2) + 10|}{\sqrt{9 + 16}}
]
3. Simplify:
[
|-3 + 8 + 10| = 15
]
4. Denominator:
[
\sqrt{25} = 5
]
Answer
[d = 3]
Question 6: Find the distance of [(p, q)] from the line [lx + my + n = 0].
Step-by-Step Solution:
1. Substitute [(x_1, y_1) = (p, q)]
2. Apply formula:
[
d = \dfrac{|lp + mq + n|}{\sqrt{l^2 + m^2}}
]
Answer:
[d = \dfrac{|lp + mq + n|}{\sqrt{l^2 + m^2}}]
Question 7: Find the value of [k] if the distance of the origin from [kx + 4y – 8 = 0] is [2].
Step-by-Step Solution:
1. Distance from origin:
[
d = \dfrac{|C|}{\sqrt{A^2 + B^2}}
]
2. Substitute:
[
\dfrac{8}{\sqrt{k^2 + 16}} = 2
]
3. Square both sides:
[
\dfrac{64}{k^2 + 16} = 4
]
4. Cross multiply:
[
64 = 4(k^2 + 16)
]
5. Simplify:
[
k^2 = 0
]
Answer:
[k = 0]
Question 8: If the perpendicular distance of [(3, 4)] from the line [ax + by – 7 = 0] is zero, find the relation between [a] and [b].
Step-by-Step Solution:
1. Distance zero ⇒ point lies on line
2. Substitute:
[
3a + 4b – 7 = 0
]
Answer:
[3a + 4b = 7]
Question 9: Find the distance between the point [(1, 1)] and the x-axis.
Step-by-Step Solution
1. Equation of x-axis:
[
y = 0 \Rightarrow 0x + y + 0 = 0
]
2. Apply formula:
[
d = \dfrac{|1|}{\sqrt{1}}
]
Answer:
[
[d = 1]
]
Question 10: Find the distance of [(a, b)] from the y-axis.
Step-by-Step Solution:
1. Equation of y-axis:
[
x = 0 \Rightarrow x + 0y + 0 = 0
]
2. Apply formula:
[
d = \dfrac{|a|}{\sqrt{1}}
]
Answer
[
[d = |a|]
]
Sign in with Google