Question

52. Find the foot of the perpendicular draw from the point p(1, 6, 3) on the line
X y-1
1 2
Z-2
3
Also find its distance from P.​

Answer 1

Step-by-step explanation:

Given Find the foot of the perpendicular drawn from the point p(1, 6, 3) on the line x/1 = y - 1/2 = z - 2/2

Also find its distance from P.​

• Let the coordinate of the foot of the perpendicular from the point P(1,6,3) on the line be Q (l1,m1,n1)
• So x/1 = y – 1 / 2 = x – 2/2------------- 1
• Since point Q lies on the line 1 we have
• So l1/1 = m1 – ½ = n1 – 2 / 2 = k
• So we get
• l1 = k, m1 = 2k + 1, n1 = 2k + 2---------2
• So the direction ratios of perpendicular line PQ will be
• a1 = k – 1
• b1 = 2k + 1 – 6
•     = 2k – 5
• c1 = 2k + 2 – 3
•      = 2k – 1
• The direction ratios of the given line will be  
• a2 = 1, b2 = 2, c2 = 2
• Since the line PQ is perpendicular to given line,
• a1a2 + b1b2 + c1c2 = 0
• So 1 (k – 1) + 2 (2k – 5) + 2(2k – 1) = 0
• So k – 1 + 4k – 10 + 4k – 2 = 0
•        9k – 13 = 0
•        9k = 13
•        k = 13/9
• Substituting the value of k in eqn 2 we get
•          l1 = 13/9
•          m1 = 2(13/9) + 1
•                = 26 / 9 + 1
•                   = 35/9
• n1 = 2(13/9) + 2
•       = 26/9 + 2
•       = 44/9
• So the coordinates of the foot of the perpendicular will be (13/9, 35/9, 44/9)
• Now distance from the point p will be
• PQ = √(1 – 13/9) ^2 + (6 – 35/9)^2 + (3 – 44/9)^2
•      = √ (-4/9)^2 + (19/9)^2 + (-17/9)^2
•      = √16/81 + 361/81 + 289/81
•      = √666 / 81
•      = 25.8 / 9
•      = 2.866                                          

Reference link will be

https://brainly.in/question/16836084