Examine which of the following conicoids are central and which are non-central. Also determine which of the central conicoids have centre at the origin.
(i) 𝑥2 + 𝑦2 + 𝑧2 + 4𝑥 + 3𝑦 − 𝑧 = 0
(ii) 2𝑥2 − 𝑦2 − 𝑧2 + 𝑥𝑦 + 𝑦𝑧 − 𝑧𝑥 = 1
(iii) 𝑥2 + 𝑦2 − 𝑧2 − 2𝑥𝑦 − 3𝑦𝑧 − 6𝑧𝑥 + 𝑥 − 2𝑦 + 5𝑧 + 4 = 0
A conicoid given by equation
"ax^2 +by^2 +cz^2 +2fyz+2gxz+2hxy"has a point "(x_0 ,y_0 ,z_0 )" as a center if
(i)
"a=1, b=1, c=1,"
"f=g=h=0,"
"u=2,v=3\/2, w=-1\/2"
Then
The central conicoid has a center "(-2,-3\/2,1\/2)."
(ii)
"a=2, b=-1, c=-1,"
"f=1\/2,g=-1\/2,h=1\/2,"
"u=v=w=0"
Then
The central conicoid has a center "(0,0,0)."
(iii)
"a=1, b=1, c=-1,"
"f=-3\/2,g=-3,h=-1,"
"u=1\/2,v=-1,w=5\/2"
Then
The central conicoid has a center "(\\dfrac{49}{162},\\dfrac{92}{81},-\\dfrac{1}{9})."
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