Consider the system of equations a1x-b1y-c1z-0- a2x-b2y-c2z-0- a3x-b3y-c3z-0-If - a1 b1 c1 a2 b2 c2 a3 b3 c3 -0- then the system has

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General Equation of Ellipse

Consider the ...

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Consider the system of equations $a_{1} x + b_{1} y + c_{1} z = 0,$ $a_{2} x + b_{2} y + c_{2} z = 0,$ $a_{3} x + b_{3} y + c_{3} z = 0,$ If $∣ ∣ ∣ ∣ \begin{matrix} a_{1} & b_{1} & c_{1} a_{2} & b_{2} & c_{2} a_{3} & b_{3} & c_{3} \end{matrix} ∣ ∣ ∣ ∣ = 0$ , then the system has

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|\begin{matrix} a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \\ a_{3} & b_{3} & c_{3} \end{matrix}|

A = ∣ ∣ ∣ ∣ \begin{matrix} a_{1} & b_{1} & c_{1} a_{2} & b_{2} & c_{2} a_{3} & b_{3} & a_{3} \end{matrix} ∣ ∣ ∣ ∣ \neq 0,

a_{1} x + b_{1} y + c_{1} z = 0, a_{2} x + b_{2} y + c_{2} z = 0

a_{3} x + b_{3} y + c_{3} z = 0

A = ⎡ ⎢ ⎣ \begin{matrix} a_{1} & b_{1} & c_{1} a_{2} & b_{2} & c_{2} a_{3} & b_{3} & c_{3} \end{matrix} ⎤ ⎥ ⎦

| A | \neq 0

a_{1} x + b_{1} y + c_{1} z = 0, a_{2} x + b_{2} y + c_{2} z = 0

a_{3} x + b_{3} y + c_{3} z = 0

b c + q r = c a + r p = a b + p q = - 1

∣ ∣ ∣ ∣ \begin{matrix} a p & a & p b q & b & q c r & c & r \end{matrix} ∣ ∣ ∣ ∣ = 0 (a b c, p q r \neq 0)

a_{1} x + b_{1} y + c_{1} = 0, a_{2} x + b_{2} y + c_{2} = 0, a_{3} x + b_{3} y + c_{3} = 0

∣ ∣ ∣ ∣ \begin{matrix} a_{1} & b_{1} & c_{1} a_{2} & b_{2} & c_{2} a_{3} & b_{3} & c_{3} \end{matrix} ∣ ∣ ∣ ∣ = 0

a_{1} x + b_{1} y + c_{1} z = 0 . . . (1),

a_{2} x + b_{2} y + c_{2} z = 0 . . . (2),

a_{3} x + b_{3} y + c_{3} z = 0 . . . (3)

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