Question

Y+z-x=k,(z+x-y)(x+y-z) is proportional to yz then prove that x+y+z is proportional to yz

Answer 1

Proof:

Given, y + z - x = k ..... (1)

Also (z + x - y) (x + y - z) ∝ yz

or, (z + x - y) (x + y - z) = cyz ,

where c is proportional constant

or, xz + yz - z² + x² + xy - xz - xy - y² + yz = cyz

or, x² - y² - z² + 2yz = cyz

or, x² - (y² + 2yz + z²) = cyz - 2yz - 2yz

or, x² - (y + z)² = (c - 4) yz

or, (x - y - z) (x + y + z) = (c - 4) yz

or, {- (y + z - x)} (x + y + z) = (c - 4) yz

or, - k (x + y + z) = (c - 4) yz, by (1)

or, x + y + z = (4 - c)/k * yz

Since (4 - c)/k is a constant value, we can write

x + y + z ∝ yz

Hence, proved.

Answer 2

Answer:

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