Question

1/a^2 +1/b^2 +1/c^2 =1/ab +1/bc +1/ca then p.t a=b=c

Answer 1

let

$\frac{1}{a} = x \\ \frac{1}{b} = y \\ \frac{1}{c} = z$

then, according to the given condition

x² + y² + z² = xy + yz + zx

x² + y² + z² - (xy + yz + zx) = 0

multiplying throughout by 2

2x² + 2y² + 2z² - 2xy - 2yz - 2zx = 0

(x² - 2xy + y²) + (y² - 2yz + z²) + (z² - 2zx + x²) = 0

(x - y)² + (y - z)² + (z - x)² = 0

then,

(x - y)² = 0

(y - z)² = 0

(z - x)² = 0

x - y = 0

y - z = 0

z - x = 0

hence,

x = y

y = z

z = x

that is,

x = y = z

$\frac{1}{a} = \frac{1}{b} = \frac{1}{c}$

using invertendo,

a = b = c

... Hence Proved!