Canonical Forms
We will cover following topics
Reduction To Canonical Forms
The equation for a conicoid can be reduced to its canoninal (normal) form by change of coordinate system. Some examples of conicoids and their canonical forms are given below.
Ellipsoid
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Ellipsoid: \(\dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}+\dfrac{z^{2}}{c^{2}}-1=0\)
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Imaginary Ellipsoid: \(\dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}+\dfrac{z^{2}}{c^{2}}+1=0\)
Hyperboloid
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Hyperboloid of one sheet: \(\dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}-\dfrac{z^{2}}{c^{2}}-1=0\)
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Hyperboloid of two sheets: \(\dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}-\dfrac{z^{2}}{c^{2}}+1=0\)
Cone
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Second order Cone: \(\dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}-\dfrac{z^{2}}{c^{2}}=0\)
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Imaginary second degree Cone: \(\dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}+\dfrac{z^{2}}{c^{2}}=0\)
Paraboloid
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Elliptic Paraboloid: \(\dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}-2 c z=0\)
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Hyperbolic Paraboloid: \(\dfrac{x^{2}}{a^{2}}-\dfrac{y^{2}}{b^{2}}-2 c z=0\)
PYQS
Reduction to Canonical Forms
1) Reduce the following equation to the standard form and hence determine the conicoid:
\(x^{2}+y^{2}+z^{2}\)-\(y z-z x-x y-3 x-6 y\)-\(9 z+21=0\)
[2017, 15M]
2) Reduce the following equation to canonical form and determine which surface is represented by it:
\(x^{2}-7 y^{2}+2 z^{2}\)-\(10 y z-8 z x-10 x y\)+\(6 x+12 y-6 z+2=0\)
[2005, 15M]
3) Show that the equation \(9x^2-16y^2-18x-32y-15=0\) represents a hyperbola. Obtain its eccentricity and foci.
[2002, 12M]
4) Find the equation of the cubic curve which has the same asymptotes as
\[2x(y-3)^3=3y(x-1)^2\]and which touches the \(x-axis\) at the origin and passes through the point (1,1).
[2001, 15M]
5) Find the equation of the circle circumscribing the triangle formed by the points \((a,0,0)\), \((0,b,0)\), \((0,0,c)\). Obtain also the coordinates of the centre of the circle.
[2001, 15M]