Second Degree Equations In Three Variables

The general equation of the second degree in in three variables is given by $\mathrm{f}(\mathrm{x}, \mathrm{y}, \mathrm{z})=\mathrm{ax}^{2}+\mathrm{by}^{2}+\mathrm{cz}^{2}+2 \mathrm{fyz}+2 \mathrm{gxz}+2 \mathrm{hxy}+2 \mathrm{px}+2 \mathrm{qy}+2 \mathrm{rz}+\mathrm{d}=0$.

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

Hyperboloid

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

Cone

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

Paraboloid

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

Cylinder

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

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 $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2}=1$, $z=0$ to meet two fixed lines $y=mx$, $z=c$ and $y=-mx$, $$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 $ax^2+by^2=1$ intercepts a constant distance $2k$ on the $y-axis$. Prove that the locus of their point of intersection is the conic

\[ax^2(ax^2+by^2-1) = bk^2(ax^2-1)^2\]

[2006, 12M]

7) If $PSP'$ and $QSQ'$ are the two perpendicular focal chords of a conic $\dfrac{1}{r}=1+ e \cos \theta$, prove that

\[\dfrac{1}{SP.SP'} + \dfrac{1}{SQ.SQ'}\]

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 $y^2=4ax$.

[2004, 15M]

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

[2004, 12M]

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