Second Degree Equations In Three Variables

The general equation of the second degree in in three variables is given by $f (x, y, z) = {a x}^{2} + {b y}^{2} + {c z}^{2} + 2 f y z + 2 g x z + 2 h x y + 2 p x + 2 q y + 2 r z + d = 0$ .

This equation describes a quadric surface, also known as a conicoid.

Some examples of conicoids are ellipsoids, hyperboloids and paraboloids.

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

Second Degree Equations in Three Variables

1) Find the equation of the tangent at the point (1,1,1) to the conicoid $3 x^{2} - y^{2} = 2 z$ .

[2017, 10M]

2) Find the locus of the points of intersection of three mutually perpendicular tangent planes to $a x^{2} + b y^{2} + c z^{2} = 1$ .

[2017, 10M]

3) Find the locus of the point of intersection of three mutually perpendicular tangent planes to the conicoid $a x^{2} + b y^{2} + c z^{2} = 1$ .

[2016, 15M]

4) A line is drawn through a variable point on the ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ , $z = 0$ to meet two fixed lines $y = m x$ , $z = c$ and $y = - m x$ , $$ z = - c$ . Find the locus of the line.

[2009, 12M]

5) Find the locus of the point which moves so that its distance from the plane $x + y - z = 1$ is twice its distance from the line $x = - y = z$ .

[2007, 12M]

6) A pair of tangents to the conic $a x^{2} + b y^{2} = 1$ intercepts a constant distance $2 k$ on the $y - a x i s$ . Prove that the locus of their point of intersection is the conic

a x^{2} (a x^{2} + b y^{2} - 1) = b k^{2} (a x^{2} - 1)^{2}

[2006, 12M]

7) If $P S P^{'}$ and $Q S Q^{'}$ are the two perpendicular focal chords of a conic $\frac{1}{r} = 1 + e \cos θ$ , prove that

\frac{1}{S P . S P^{'}} + \frac{1}{S Q . S Q^{'}}

is constant.

[2006, 15M]

8) Find the locus of the middle points of the chords of the rectangular hyperbola $x^{2} - y^{2} = a^{2}$ which touch the parabola $y^{2} = 4 a x$ .

[2004, 15M]

9) Prove that the locus of the foot of the perpendicular drawn from the vertex on a tangent to the parabola $y^{2} = 4 a x$ is $(x + a) y^{2} + x^{3} = 0$ .

[2004, 12M]

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