Question

The length of the perpendicular from the point (2, –1, 4) on the straight line, (x + 3)/10 = (y - 2)/-7 = z/1
is:
(A) greater than 2 but less than 3 (B) less than 2
(C) greater than 4 (D) greater than 3 but less than 4

Answer 1

The length of the perpendicular from the point (2, –1, 4) on the straight line, (x + 3)/10 = (y - 2)/-7 = z/1  is greater than 3 but less than 4.

Let (2,-1,4) be point P

• Let (x + 3)/10 = (y - 2)/-7 = z/1  = a , a constant
• Vector along the line is L = 10i -7j + k
• ==> x = 10a -3 , y = -7a + 2, z =a
• Let (x,y,z) , be point A, and is the foot of the perpendicular from (2,-1,4) on the line.
• Now lets solve for a.
• Vector along the perpendicular , PA = (2,-1,4) - (x,y,x).
• PA = (2 - (10a-3))i + (-1 - (-7a +2))j + (4 - a)k = (5 -10a)i + (7a -3)j + (4 - a)k
• Since PA is perpendicular to the line,
• PA.L = 0

• ((5 -10a)i + (7a -3)j + (4 - a)k ).10i -7j + k = 50 - 100a -49a +21 + 4 -a = 0
• 75 = 150a ==>
• a = 1/2

Now point A is (2, -3/2, 1/2)

• Vector PA =  0i + 0.5j + 3.5k

• DIstance PA = |PA| = $\sqrt{0^{2} + 0.5^{2} + 3.5^{2} }$ =  $\sqrt{0.25 + 12.25} = \sqrt{12.5} = \sqrt{\frac{50}{4} } = \frac{5}{\sqrt{2} }$

• $\frac{5}{\sqrt{2} }$ = 3.53

Therefore the length of perpendicular is greater than 3 but less than 4.

Answer is option (c)