Distance Between Lines - Free MCQ Practice Test with solutions, JEE Maths

• The given equation is 
• x +√3y = 2 
• On dividing both sides by  
• [(−1)² + (√3)²]^1/2 = √4 = 2
•  = x/2 + √3y/2 = 2/2 
• Therefore , the equation becomes
• = (1/2)x + (√3/2)y = 1
• = x cos 60° + y sin 60° = 1
• Normal form of any line is : x cosw + y sinw = p
• w = 60°

The foot of perpendicular can be found by equating the distance between the two points and the distance between point and line. This is found to be (68/25,-49/25).

Step 1: Simplify the line equation
2(x – 3) = 3(y + 5)
⇒ 2x – 6 = 3y + 15
⇒ 2x – 3y – 21 = 0

So, the line equation is:
2x – 3y – 21 = 0

Step 2: Formula for distance of a point from a line
Distance = |Ax + By + C| ÷ √(A² + B²)
Where the line is Ax + By + C = 0 and point is (x, y).

Here: A = 2, B = –3, C = –21, and point = (3, 3).

Step 3: Substitute values
Numerator = |(2)(3) + (–3)(3) + (–21)|
= |6 – 9 – 21|
= |–24|
= 24

Denominator = √(2² + (–3)²) = √(4 + 9) = √13

So, distance = 24/√13

Given lines:
3(x+y)+2 = 0
⇒ 3x+3y+2 = 0
6x+6y+28 = 0
⇒ 3x+3y+14 = 0 (divide by 2)

They are parallel since coefficients of x and y are the same.

Distance between parallel lines ax+by+c₁ = 0 and ax+by+c₂ = 0 is

In Δ PAM & Δ PAN, angle (PAM) = angle (PAN) (Since AP bisects the angle (BAC)) angle (AMP) = angle (ANP) = 900
PA = PA (common)
By AAS congruence, Δ PAM and Δ PAN are congruent. 
MP = NP = 2 cm

√3x + y - 8
√3x + y = 8
Dividing by √[(√3)2 + (1)2]
= √[3+1]
= √4
= 2
√3x/2 + y/2 = 8/2
√3x/2 + y/2 = 4
x(√3/2) + y(1/2) = 4.....(1)
Normal form of any line : xcos w + ysin w = p....(2)
Comparing (1) and (2)
p = 4

4x - 3y + 5 = 0,  4x - 3y + 15 = 0
A = 4,  B = -3   c₁ = 5,   c₂ = 15
|c₁ - c₂|/[A² + B²]^1/2
= |5 - 15|/[(4)² + (-3)²]^½
= 10/5
= 2