?
If x = b + c - 2a, y = c + a - 2b, z = a + b - 2c, then the value of x2 + y2 - z2 + 2xy is?
Answer & Solution
Correct Answer:
Option
A
x = b + c - 2a
y = c + a - 2b
z = a + b - 2c
⇒ x + y + z = (b + c - 2a) + (c + a - 2b) + (a + b - 2c) = 0
∴ Now,
⇒ x2 + y2 + 2xy - z2
⇒ (x + y)2 - z2 [∴ (a2 - b2) = (a + b) (a - b)]
⇒ (x + y - z) (x + y + z)
As we know, (x + y + z) = 0
∴ x2 + y2 - z2 + 2xy
= 0 × (x + y - z)
= 0
y = c + a - 2b
z = a + b - 2c
⇒ x + y + z = (b + c - 2a) + (c + a - 2b) + (a + b - 2c) = 0
∴ Now,
⇒ x2 + y2 + 2xy - z2
⇒ (x + y)2 - z2 [∴ (a2 - b2) = (a + b) (a - b)]
⇒ (x + y - z) (x + y + z)
As we know, (x + y + z) = 0
∴ x2 + y2 - z2 + 2xy
= 0 × (x + y - z)
= 0
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