If a2 + b2 + c2 = 16, x2 + y2 + z2 = 25 and ax + by + cz

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If a² + b² + c² = 16, x² + y² + z² = 25 and ax + by + cz = 20, then the value of a + b + c is
x + y + z

1. 3
  5
2. 5
  3
3. 4
  5
4. 5
  4

Correct Option: C

(ax + by + cz)² = (a² + b² + c²)(x² + y² + z²) = 400
⇒ a²x² + b²y² + c²z² + 2abxy + 2bcyz + 2acxz
= a²x² + a²y² + a²z² + b²x² + b²y² + b²z² +
c²x² + c²y² + c²z²
⇒ a²y² + a²z² + b²x² + b²z² +
c²x² + c²y² = 2abxy + 2bcyz + 2acxz
⇒ a²y² - 2abxy + b²x² + a²z² + c²x² - 2acxz + b²z² + c²y² - 2bcyz = 0
⇒ (ay – bx)² + (az – cx)² + (bz – cy)² = 0

⇒ ay – bx = 0 ⇒ ay = bx ⇒	a	=	x
	b		y

az – cx = 0 ⇒ az = cx ⇒	a	=	x
	c		z

∴ a = kx ; b = ky; c = kz
∴ a² + b² + c² = 16
⇒ k²(x² + y² + z²) = 16
⇒ k² × 25 = 16

⇒ k² =	16	⇒ k =	4
	25		5

∴	a + b + c	= k =	4
	x + y + z		5

If a² + b² + c² = 16, x² + y² + z² = 25 and ax + by + cz = 20, then the value of	a + b + c	is
	x + y + z