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Question:

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

Explanation:

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).