CAT 2019 Question Paper | Quants Slot 2

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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})

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Video Explanation

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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

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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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