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

Ellipsoid: $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} + \frac{z^{2}}{c^{2}} - 1 = 0$
Imaginary Ellipsoid: $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} + \frac{z^{2}}{c^{2}} + 1 = 0$

Hyperboloid

Hyperboloid of one sheet: $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} - \frac{z^{2}}{c^{2}} - 1 = 0$
Hyperboloid of two sheets: $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} - \frac{z^{2}}{c^{2}} + 1 = 0$

Cone

Second order Cone: $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} - \frac{z^{2}}{c^{2}} = 0$
Imaginary second degree Cone: $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} + \frac{z^{2}}{c^{2}} = 0$

Paraboloid

Elliptic Paraboloid: $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} - 2 c z = 0$
Hyperbolic Paraboloid: $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} - 2 c z = 0$

Cylinder

Elliptic Cylinder: $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} - 1 = 0$
Imaginary Elliptic Cylinder: $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} + 1 = 0$
Parabolic Cylinder: $y^{2} - 2 p x = 0$
Hyperbolic Cylinder: $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} - 1 = 0$

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 $9 x^{2} - 16 y^{2} - 18 x - 32 y - 15 = 0$ represents a hyperbola. Obtain its eccentricity and foci.

[2002, 12M]

4) Find the equation of the cubic curve which has the same asymptotes as

2 x (y - 3)^{3} = 3 y (x - 1)^{2}

and which touches the $x - a x i s$ 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]

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