Test: Distance of a Point from a Line (7 July) - JEE MCQ

Perpendicular Distance of a Point from a Line:
Let us consider a plane given by the Cartesian equation, Ax + By + C = 0
And a point whose coordinate is, (x₁, y₁)
Now, distance = 
Calculation:
Given equation of line is 3x cos ϕ + 5y sin ϕ = 15
⇒ 3x cos ϕ + 5y sin ϕ – 15 = 0
Let perpendiculars distance from the two points (±4, 0) are p₁ and p₂
Lengths of perpendicular from the point (+ 4, 0),

Lengths of perpendicular from the point (- 4, 0),

Multiplying equation 1 and 2,

Concept:
The distance of a point (h,k) from a line ax + by + c = 0 is given by,

Calculation:
Solving the lines 2x - 3y + 5 = 0 and 3x + 4y = 0 

⇒ The point of intersection is 
⇒ The distance of the point  from the line 5x - 2y = 0 is

The perpendicular distance d from P (x₁, y₁) to the line ax + by + c = 0 is given by  
CALCULATION:
Given :The length of the perpendicular from the point (4, 1) on the line 3x - 4y + k = 0 is 2 units.
Let P = (4, 1)
Here x₁ = 4, y₁ = 1, a = 3, b = - 4 and d = 2
Now substitute x₁ = 4 and y₁ = 1 in the equation 3x - 4y + k = 0
⇒ |3⋅ x₁ - 4 ⋅ y₁ + k| = |12 - 4 + k| = |8 + k|

As we know that, the perpendicular distance d from P (x₁, y₁) to the line ax + by + c = 0 is given by 

⇒ |8 + k| = 10
⇒ k = 2 or - 18
Hence, option C is the correct answer.

Concept:
Distance formula-
Distance between two points (x₁,y₁) and (x₂,y₂) = 
Calculation:
On y-axis, x coordinate is 0
Let the point on the y-axis be be (0,a)
⇒ (0,a) is equidistant from the points (3,2) and (-1,3) 

Squaring both sides,

⇒ 2a = −3  or a = -3/2
⇒ Point is (0, -3/2)
∴ The correct answer is option (2).

Given Data:
First equation: 3x + 2y = 5
Second equation: 2x - 3y = 7
Concept or Formula:
The consistency of a pair of lines can be checked using their coefficients. If the lines are represented as a₁x + b₁y = c₁ and a₂x + b₂y = c₂, the lines are consistent (have a mutual point of intersection) if a₁/a₂ ≠ b₁/b₂. If a₁/a₂ = b₁/b₂ ≠ c₁/c₂, the lines are inconsistent (parallel & never intersect).
Solution:
Compare the coefficients of the two equations.
⇒ The ratio of the coefficients of x (a₁/a₂) = 3/2
⇒ The ratio of the coefficients of y (b₁/b₂) = 2/(-3) = -2/3
Since 3/2 ≠ -2/3, the lines are consistent.
Therefore, the pair of lines represented by the equations 3x + 2y = 5 and 2x - 3y = 7 are consistent.

Concept:
The distance of a point (x₁, y₁) from a line ax + by + c = 0 

Calculation:
Given line 3y = 4x + 5
⇒ 4x - 3y + 5 = 0
(x₁, y₁) = (2, 1)

Concept:
Dot product of two perpendicular lines is zero.
Distance between two points (x₁, y₁, z₁) and (x_2, y₂, z₂) is given by, 

Calculation:
Let M be the foot of perpendicular drawn from the point P(2, 3, 4)

x = k, y  = 0, z = 0
So M = (k, 0, 0)
Now direction ratios of PM = (2 - k, 3 - 0, 4 - 0) = (2- k, 3, 4) and direction ratios of given line are 1, 0, 0
PM is perpedicular to the given line so,
(2 - k) (1) + 3(0) + 4 (0) = 0
∴ k = 2
M = (2, 0, 0)
Perpendicular distance PM = 
Hence, option (2) is correct. 

Concept:
If two nonvertical lines are perpendicular, then the product of their slopes is −1.
The slope of a line passing through the distinct points (x₁, y₁) and (x₂, y₂) is 
Calculation:

Slope of line passing through points (0, k) and (3, 1)

So, the slope of line 3x - 4y - 5 = 0 is 3/4
Now since line OP and 3x - 4y - 5 = 0 are perepndicular 
 K = 5
Hence, option (3) is correct. 

CONCEPT:
The perpendicular distance d from P (x₁, y₁) to the line ax + by + c = 0 is given by  
CALCULATION:
Here, we have to find the distance of the line 3x - 4y + 12 = 0 from the point (4, 1)
Let P = (4, 1)
⇒ x₁ = 4 and y₁ = 1
Here, a = 3 and b = - 4
Now substitute  x₁ = 4 and y1 = 1 in the equation 3x - 4y + 12 = 0, we get
⇒ |3⋅ x₁ - 4 ⋅ y₁ + 12| = |12 - 4 + 12| = 20

As we know that, the perpendicular distance d from P (x₁, y₁) to the line ax + by + c = 0 is given by 

Perpendicular Distance of a Point from a Line:
Let us consider a line Ax + By + C = 0 and a point whose coordinate is (x₁, y₁) 
Calculation:
Given: equation of line is 3x + 4y = 7 and a point is (3, 2)
We know the distance of a line from is given by, 
So, distance of a point (3, 2) from a line 3x + 4y - 7 = 0 is given by,

= 10/5
= 2 units 
Hence, option (2) is correct.