Show that max- value of 2x-3y-5z 2 x 2 - y 2 - z 2 - 38

Let r⃗ ⃗_⃗ ⃗1⃗ =2î +3ĵ +5k̂ r⃗ ⃗_⃗ ⃗2⃗ =xî +yĵ +zk̂ r⃗ ⃗_⃗ ⃗1⃗r⃗ ⃗_⃗ ⃗2⃗ =| r⃗ ⃗_⃗ ⃗1⃗ #160; | | r⃗ ⃗_⃗ ⃗2⃗ #160; | cosθ #160; ⇒r⃗ ⃗_⃗ ...

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Show that max...

Question

Show that max. value of $\frac{{(2 x + 3 y + 5 z)}^{2}}{x^{2} + y^{2} + z^{2}} \leq 38$

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x^{2} + y^{2} + z^{2} = 2 (x - y - z) - 3

2 x - 3 y + 4 z

x^{2} + y^{2} + z^{2} - x y - y z - z x = \frac{1}{2} [(x - y)^{2} + (y - z)^{2} + (z - x)^{2}]

(2 x + 3 y + 5 z)^{2}

x + y + z = x y z

\frac{2 x}{1 - x^{2}} + \frac{2 y}{1 - y^{2}} + \frac{2 z}{1 - z^{2}} = \frac{2 x}{1 - x^{2}} \cdot \frac{2 y}{1 - y^{2}} \cdot \frac{2 z}{1 - z^{2}}

S = x^{2} + y^{2} + z^{2} - 2 x + 3 y - 4 z - 5 = 0

A (1, 0, - 1)

B (- 1, 0, - 2)

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