Question

if x ,y,z sre integers , y+z-x=constant and (z+x-y)(x+y-z) varies yz then prove that (x+y+z)variesyz

Answer 1

As given , x, y and z are integers.

y + z - x= constant

y + z- x = k

(z+x-y)(x+y-z) varies y z.

→(x+z-y)[x-(z-y)]= P y z

⇒ x² - (z-y)²= P y z→→→→ Using Identity A²- B²=(A-B)(A+B)

⇒x² - (z-y)²- 4 y z= P y z - 4 y z

⇒x²- z²-y²+2 y z - 4 y z=(P-4) y z

⇒x²- (y+z)²= M y z⇒⇒P-4= M

⇒(x-y-z)(x+y+z)=M y z-----(1)→→→→ Using Identity A²- B²=(A-B)(A+B)

As , y + z - x= k

x-y-z= -k= N

Putting the value of , x-y-z in equation  (1)

N (x+y+z)=M y z

x+y+z= $\frac{M}{N}$ y z

x + y + z= D y z, where $\frac{M}{N}$= D

Hence, (x+y+z)varies yz.