Math Olympiad | Can you solve given System of Equations ? | VIJAY Maths

[@Quest3669]

For real solns.
X= 8 r 0 ;y= 0 r 8
Same for other eqn .
Hence; ? = 0 By anyways

[@raghvendrasingh1289]

By Cauchy Schwartz inequality (ax+by)^2

[@abeerkale566]

Another way,
For a very generalised answer of a=b and x=y
a^2 + b^2 = 169 so a=b = 13/root2
similarly x=y=8/root2
so ax-by=0

[@key_board_x]

ax + by = 104
(ax + by)² = 104²
a²x² + 2abxy + b²y² = 104² → given: x² + y² = 64 → y² = 64 - x²
a²x² + 2abxy + b².(64 - x²) = 104² → given: a² + b² = 169 → b² = 169 - a²
a²x² + 2abxy + (169 - a²).(64 - x²) = 104²
a²x² + 2abxy + 10816 - 169x² - 64a² + a²x² = 10816
2a²x² + 2abxy - 169x² - 64a² = 0
2abxy = 64a² + 169x² - 2a²x² ← memorise this result
ay = (64a² + 169x² - 2a²x²)/2bx
ay - bx = [(64a² + 169x² - 2a²x²)/2bx] - bx
ay - bx = [64a² + 169x² - 2a²x² - (bx * 2bx)]/2bx
ay - bx = [64a² + 169x² - 2a²x² - 2b²x²]/2bx
ay - bx = [64a² + 169x² - 2x².(a² + b²)]/2bx → given: a² + b² = 169
ay - bx = (64a² + 169x² - 338x²)/2bx
ay - bx = (64a² - 169x²)/2bx ← equation (1)
Restart from the memorized result
2abxy = 64a² + 169x² - 2a²x²
bx = (64a² + 169x² - 2a²x²)/2ay
ay - bx = ay - [(64a² + 169x² - 2a²x²)/2ay]
ay - bx = [(ay * 2ay) - (64a² + 169x² - 2a²x²)]/2ay
ay - bx = [2a²y² - 64a² - 169x² + 2a²x²]/2ay → given: x² + y² = 64 → y² = 64 - x²
ay - bx = [2a².(64 - x²) - 64a² - 169x² + 2a²x²]/2ay
ay - bx = (128a² - 2a²x² - 64a² - 169x² + 2a²x²)/2ay
ay - bx = (64a² - 169x²)/2ay → recall (1): ay - bx = (64a² - 169x²)/2bx
(64a² - 169x²)/2bx = (64a² - 169x²)/2ay
2ay.(64a² - 169x²) = 2bx.(64a² - 169x²)
ay.(64a² - 169x²) = bx.(64a² - 169x²)
ay = bx
ay - bx = 0

[@SGuerra]

A questão é bonita. Parabéns pela escolha. Brasil Novembro de 2024.

[@vijaymaths5483]

Thank you so much ⚘️