If-x-cy-bz-y-cx-az-and-z-bx-ay-then-prove-that-x-1-a-2-y-1-b-2-z-1-c-2-

Question Number 214759 by MATHEMATICSAM last updated on 19/Dec/24

If x = c y + b z, y = c x + a z and

z = b x + a y then prove that

\frac{x}{\sqrt{1 - a^{2}}} = \frac{y}{\sqrt{1 - b^{2}}} = \frac{z}{\sqrt{1 - c^{2}}} .

Answered by MathematicalUser2357 last updated on 19/Dec/24

It gives us x = y = z = 0,

and \frac{0}{\sqrt{1 - a^{2}}} = \frac{0}{\sqrt{1 - b^{2}}} = \frac{0}{\sqrt{1 - c^{2}}} is true . Only for a, b, c \in (- 1, 1)

Answered by som(math1967) last updated on 19/Dec/24

x = c y + b z

\Rightarrow x = c y + b (b x + a y)

\Rightarrow x = c y + b^{2} x + a b y

\Rightarrow x (1 - b^{2}) = y (c + a b)

\Rightarrow \frac{x}{y} = \frac{(c + a b)}{(1 - b^{2})} \dots \dots . (1)

y = c x + a z

\Rightarrow y = c x + a (b x + a y)

\Rightarrow y (1 - a^{2}) = x (c + a b)

\Rightarrow \frac{x}{y} = \frac{1 - a^{2}}{c + a b} \dots \dots . . (2)

(1) \times (2)

\frac{x^{2}}{y^{2}} = \frac{c + a b}{1 - b^{2}} \times \frac{1 - a^{2}}{c + a b}

\Rightarrow \frac{x}{y} = \frac{\sqrt{1 - a^{2}}}{\sqrt{1 - b^{2}}}

∴ \frac{x}{\sqrt{1 - a^{2}}} = \frac{y}{\sqrt{1 - b^{2}}}

s a m e w a y w e c a n s h o w

\frac{y}{\sqrt{1 - b^{2}}} = \frac{z}{\sqrt{1 - c^{2}}}