Eliminate x, y, z from the equations ax + hy + gz = 0, hx + by + fz = 0, gx + fy + cz = 0.

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Solution

$a x + h y + g z = 0.... (i) h x + b y + f z = 0.... (i i) g x + f y + c z = 0..... (i i i)$
applying cross multiplication on $(i)$ and $(i i)$
$\frac{x}{h f - b g} = \frac{y}{g h - a f} = \frac{z}{a b - h^{2}} = k \Rightarrow x = k (h f - b g), y = k (g h - a f), z = k (a b - h^{2})$
substituting $x, y$ and $z$ in $(i i i)$
$g k (h f - b g) + f k (g h - a f) + c k (a b - h^{2}) = 0 f g h - b g^{2} + f g h - a f^{2} + a b c - c h^{2} = 0 a b c + 2 f g h - a f^{2} - b g^{2} - c h^{2} = 0$

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Similar questions

a x + h y + g z = 0

h x + b y + f z = 0, g x + f y + c z = λ z

λ

x, y, z, u

x = b y + c z + d u, y = c z + d u + a x, z = d u + a x + b y, u = a x + b y + c z

a x + h y + g = 0

h x + b y + f = 0,

g x + f y + c = k

\frac{1}{λ} ∣ ∣ ∣ ∣ \begin{matrix} a & h & g h & b & f g & f & c \end{matrix} ∣ ∣ ∣ ∣

λ

(1) \frac{x^{2}}{b c - f^{2}} = \frac{y^{2}}{c a - g^{2}} = \frac{z^{2}}{a b - h^{2}} .

(2) (b c - f^{2}) (c a - g^{2}) (a b - h^{2}) = (f g - c h) (g h - a f) (h f - b g) .

a x + h y + g = 0, h x + b y + f = 0, g x + f y + c = λ

λ = \frac{a b c + 2 f g h - a f^{2} - b g^{2} - c h^{2}}{a b - h^{2}}

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