If u= x/(y-z), v = y/(z-x), w=z/(x-y) solve
Answers
uv + vw + uw = -1 if u= x/(y-z), v = y/(z-x), w=z/(x-y)
Step-by-step explanation:
If u= x/(y-z), v = y/(z-x), w=z/(x-y)
then uv + vw + uw = ?
uv + vw + uw
= (x/(y - z))(y/(z-x)) + (y/(z-x))(z/(x-y)) + (x/(y-z))(z/(x-y))
= xy/(y - z)(z-x) + yz/(z-x)(x-y) + xz/(y-z)(x-y)
= (xy(x - y) + yz(y-z) + xz(z - x)) / ((x-y)(y-z)(z-x) )
Let solve numerator only
xy(x - y) + yz(y-z) + xz(z - x)
= xy(x - y) + y²z - yz² + xz² - x²z
= xy(x - y) + z²(x - y) -z(x² - y²)
= xy(x - y) + z²(x - y) -z(x + y)(x - y)
= (x - y) ( xy + z² - z(x + y))
= (x - y) ( xy - zx + z² - zy)
=(x - y) (x(y - z) - z(y - z) )
= (x - y) (x - z)(y - z)
= (x - y )(y - z)( -(z - x))
= - (x-y)(y-z)(z-x)
=> uv + vw + uw = - (x-y)(y-z)(z-x) / ((x-y)(y-z)(z-x) )
=> uv + vw + uw = -1
Learn More :
if u=xy+yz+zx,v=x²+y²+z² and w=x+y+z
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