If a = (3,4) and B is a variable point on the lines |x| = 6. If AB ≤ 4 , then the no. of positions of B with integral coordinates is :-(a) 5 (b) 6 (c) 10 (d) 12

|x| = 6

 x can be 6 or -6

let point be B = { (6,y) or (-6,y) }

AB can be

 AB = [(6-3)² + (y-4)²]^1/2        or      [(-6-3)²+(y-4)²]^1/2 

AB ≤ 4  so

  9 + (y-4)² ≤ 16                          or      81 + (y-4)² ≤ 16

(y-4)² - (root7)² ≤0    or     ( minimum value of (y-4)² =0, enequality does not holds good)

only possible solution is 

(y-4)²-(root7) ²  ≤ 0

  (y-(4+root7)) (y-(4-root7))  ≤ 0

 for this enequality ,  4-root7 ≤  y   ≤ 4 + root7

                                         1.36  ≤  y  ≤ 6.64

y can take value 2 , 3, 4, 5, 6

total solutions for y are 5 , total number of points should be 5...

so option a is correct...

approve if u like my ans         

Approved