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Finding the Value of an Expression

If x2+y2+z2...

Question

If $x^{2} + y^{2} + z^{2} = 2 (x - y - z) - 3$ , find the value of $2 x - 3 y + 4 z$ .

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Solution

Given, $x^{2} + y^{2} + z^{2} = 2 (x - y - z) - 3$
$\Rightarrow (x^{2} - 2 x) + (y^{2} + 2 y) + (z^{2} + 2 z) + 3 = 0$
$\Rightarrow (x^{2} - 2 x + 1) + (y^{2} + 2 y + 1) + (z^{2} + 2 z + 1) = 0$
$\Rightarrow (x - 1)^{2} + (y + 1)^{2} + (z + 1)^{2} = 0$
$\Rightarrow (x - 1) = 0$ and $(y + 1) = 0$ and $(z + 1) = 0$
$\Rightarrow x = 1$ and $y = - 1$ and $z = - 1$
$∴ 2 x - 3 y + 4 z = 2 + 3 - 4 = 1$

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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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\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}}

Q. The value of

x, y

z

respectively on simplifying the equation

2 x + 3 y = 0

3 y + 4 z = 14

2 x + 4 z = 26

Q. The value of

x, y

z

respectively, on simplifying the equations

2 x + 3 y = 0, 3 y + 4 z = 14, 2 x + 4 z = 26

Q. The intersection of the sphere

x^{2} + y^{2} + z^{2} + 7 x - 2 y - z = 13

x^{2} + y^{2} + z^{2} - 3 x + 3 y + 4_{Z} = 8

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