# CAT 2019 Question Paper | Quants Slot 2

###### CAT Previous Year Paper | CAT Quants Questions | Question 3

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Question 3 : 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

1. k = 0
2. k > $$frac{5}{13}$ 3. k = $\frac{5}{13}$ 4. 0 < k ≤ $\frac{5}{13}$ ## 🚀 Introductory Offer! Buy CAT'21 95-99 Booster Course for just Rs. 4500 Valid until 19th Sep #### 2IIM : Best Online CAT Coaching. ### Video Explanation ## Best CAT Coaching in Chennai #### CAT Coaching in Chennai - CAT 2021Online Batches Available Now! ### Explanatory Answer Given equations are a2 + b2 = 25 and x2 + y2 = 169 We know 52 = 25 and 132 = 169 Multiply both equations to get$a2 + b2) (x2 + y2) = 25 x 169
(a2 + b2) (x2 + y2) = 6225
We know, 6225 = 652
We also know that ax + by = 65
So, numerically (Not algebraically), (a2 + b2) (x2 + y2) = (ax + by)2
(a2 + b2) (x2 + y2) = (ax + by)2
Expanding the equation,
a2 x2 + a2 y2 + b2 x2 + b2 y2 = a2 x2 + b2 y2 + 2axby
a2 y2 + b2 x2 = 2axby
a2 y2 + b2 x2 - 2axby = 0
This is of the form, (a-b)2
(ay-bx)2 = 0
ay - bx = 0
K = 0

The question is "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 "

##### Hence, the answer is k = 0

Choice A is the correct answer.

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