CAT 2019 Slot 2 — QA Question 3
Number TheoryEasy
Let a, b, x, y be real numbers such that a2 + b2 = 25 , x2 + y2 = 169 and ax + by = 65. If k = ay - bx, then
Answer & solution
k = 0
- B
k > 513
- C
k = 513
- D
0 < k ≤ 513
Solution
Given: a2 + b2 = 25 and x2 + y2 = 169
We know 52 = 25 and 132 = 169
Multiply both equations to get (a2 + b2) (x2 + y2) = 25 × 169
(a2 + b2) × (x2 + y2) = 4225
We know, 4225 = 652
We also know that ax + by = 65
So, numerically (Not algebraically),
(a2 + b2) × (x2 + y2) = (ax + by)2
Expanding the equation,
⇒ (ax)2 + (ay)2 + (bx)2 + (by)2 = (ax)2 + (by)2 + 2axby
⇒ (ay)2 + (bx)2 = 2axby
⇒ (ay)2 + (bx)2 - 2axby = 0
This is of the form, (p - q)2
(ay - bx)2 = 0
⇒ ay - bx = 0 = k
Hence, option (a).