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Parametric Equation of Normal

Prove that ...

Question

Prove that $Δ = ∣ ∣ ∣ ∣ \begin{matrix} a x - b y - c z & a y + b x & c x + a z a y + b x & b y - c z - a x & b z + c y c x + a z & b z + c y & c z - a x - b y \end{matrix} ∣ ∣ ∣ ∣$
$= (x^{2} + y^{2} + z^{2}) (a^{2} {+ b}^{2} {+ c}^{2}) (a x + b y + c z) .$

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Solution

Operations are suggested by the form of factors required.
Multiply $C_{1}$ by $x$ and hence divide by $x$ and then apply
$C_{1} + (y C_{2} + z C_{3})$ , we get
$Δ = \frac{1}{x} ⎡ ⎢ ⎢ ⎣ \begin{matrix} a (x^{2} + y^{2} + z^{2}) & a y + b x & c x + a z b (x^{2} + y^{2} + z^{2}) & b y - c z - a x & b z + c y c (x^{2} + y^{2} + z^{2}) & b z + c y & c z - a x - b y \end{matrix} ⎤ ⎥ ⎥ ⎦$
Take $\sum x^{2}$ from $C_{1}$ and then again multiply $R_{1}$ by a and divide $Δ$ by a Now operate
$R_{1} + (b R_{2} + c R_{3})$
$Δ = \frac{1}{a x} (x^{2} + y^{2} + z^{2})$
$⎡ ⎢ ⎢ ⎢ ⎣ \begin{matrix} a^{2} + b^{2} + c^{2} & y (a^{2} + b^{2} + c^{2}) & z (a^{2} + b^{2} + c^{2}) b & b y - c z - a x & b z + c y c & b z + c y & c z - a x - b y \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎦$
$= \frac{1}{a x} (x^{2} + y^{2} + z^{2}) (a^{2} + b^{2} + c^{2})$
$\times ⎡ ⎢ ⎣ \begin{matrix} 1 & y & z b & b y - c z - a x & b z + c y c & b z + c y & c z - a x - b y \end{matrix} ⎤ ⎥ ⎦$
Now make two zeros by applying $R_{2} - b R_{1}$ and $R_{3} - C R_{1}$
$∴ Δ = ⎡ ⎢ ⎣ \begin{matrix} 1 & y & z 0 & - (c z - a x) & c y 0 & b z & - (a x - b y) \end{matrix} ⎤ ⎥ ⎦$
$= (c z + a x) (a x + b y) - b c y z$
$= a x (a x + b y + c z)$ $∴ Δ = \sum x^{2} . \sum a^{2} . \sum a x$

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Similar questions

a^{2} + b^{2} + c^{2} = 1

x^{2} + y^{2} + z^{2} = 1

a x + b y + c z < 1

a^{2} + b^{2} + c^{2} = 1

x^{2} + y^{2} + z^{2} = 1

where a, b, c, x, y, z are real, prove that

a x + b y + c z \leq 1

a^{x} = b^{y} = c^{z}

b^{2} = a c

y = \frac{2 x z}{x + z}

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

then prove that

\frac{x}{a} = \frac{y}{b} = \frac{z}{c}

\frac{a x}{y} + \frac{b y}{x} = c, \frac{b y}{z} + \frac{c z}{y} = a, \frac{c z}{x} + \frac{a x}{z} = b,

then find the value of

\frac{y z}{x^{2}} + \frac{z x}{y^{2}} + \frac{x y}{z^{2}}

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