5. Three-Dimensional Geometry

Cone

A cone is a surface generated by a straight line that passes through a fixed point (vertex) and intersects a given fixed curve (guide curve).

For: UP PGT Mathematics UP TGT Mathematics DSSSB PGT / TGT KVS / NVS PGT LT Grade

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Definition and statement

General Equation of a Cone with Vertex at the Origin

A homogeneous equation of second degree in three variables, $F(x, y, z) = ax^2 + by^2 + cz^2 + 2fyz + 2gzx + 2hxy = 0$, represents a cone whose vertex is at the origin $(0, 0, 0)$.

📐 Geometric Interpretation

Every point on the surface lies on a line connecting the origin to the guiding curve. Scaling the coordinates by a scalar $k$, i.e., $F(kx, ky, kz) = k^2 F(x, y, z) = 0$, confirms homogeneity and guarantees that the entire ray from the origin lies on the surface.

⚙️ Algebraic & Structural Meaning

Homogeneity of degree $n$ ensures that any line passing through the origin intersects the surface at multiples of the same directional ratios, making the second-degree homogeneous equation the foundational algebraic model for central cones.

Conditions and hypotheses

Check each condition and what changes when it is omitted:

Hypothesis / Condition	Domain / Interval	Necessity	What Fails If Condition Dropped
Homogeneity of Degree Two	`Algebraic degree condition`	Strictly Mandatory	If non-homogeneous terms of degree 0 or 1 are present without shifting the origin, the vertex is not at $(0, 0, 0)$.
Vertex Specification	`Fixed point $(alpha, \beta, \gamma)$`	Strictly Mandatory	Shifting the vertex requires transforming coordinates via $X = x - \alpha$, $Y = y - \beta$, $Z = z - \gamma$ before applying homogeneity.
Real Generators Condition	`Discriminant of quadratic form $\Delta \neq 0$`	Essential for non-degenerate cones	If the discriminant vanishes, the cone degenerates into a pair of intersecting planes.

Useful results and shortcuts

Use these identities and results when their conditions apply:

Condition for Three Mutually Perpendicular Generators

$$a + b + c = 0$$

Rule: For the general cone $ax^2 + by^2 + cz^2 + 2fyz + 2gzx + 2hxy = 0$ to possess three mutually perpendicular generators, the sum of the coefficients of $x^2, y^2, z^2$ must equal zero.

Equation of Tangent Plane

$$axx_1 + byy_1 + czz_1 + f(yz_1 + zy_1) + g(xz_1 + zx_1) + h(xy_1 + yx_1) = 0$$

Rule: Replace $x^2$ by $xx_1$, $y^2$ by $yy_1$, $z^2$ by $zz_1$, $2xy$ by $xy_1 + yx_1$, etc., to write the tangent plane at $(x_1, y_1, z_1)$.

Angle Between the Lines of Intersection

$$\tan \theta = \frac{2 \sqrt{h^2 - ab}}{a + b}$$

Rule: For a cone intersecting a plane $z = 0$ or via homogeneous cross-sections, the semi-vertical angle or angle between generators uses this standard planar reduction.

Application: Direct formula substitution tested in UP TGT.

Reciprocal Cone Condition

$$A x^2 + B y^2 + C z^2 + 2F yz + 2G zx + 2H xy = 0$$

Rule: The reciprocal cone's coefficients are the cofactors $A, B, C, F, G, H$ of the original coefficient matrix.

Application: Advanced PGT question type for verifying reciprocal relations.

Enveloping Cone Vertex Condition

$$S(x_1, y_1, z_1) = 0$$

Rule: When finding the enveloping cone from a point to a sphere or conicoid, the vertex coordinates satisfy the tangent reciprocity relation.

Application: Tested in UP PGT to find apex coordinates given guiding curves.

Right Circular Cone Semi-Vertical Angle

$$\cos \alpha = \frac{la + mb + nc}{\sqrt{l^2 + m^2 + n^2}\sqrt{a^2 + b^2 + c^2}}$$

Rule: Use the dot product between the axis direction ratios $(l, m, n)$ and generator direction ratios $(a, b, c)$.

Common errors and counterexamples

These examples show how a conclusion can fail when a condition is missing:

⚠️ COMMON ERROR #01

Confusing Vertex at Origin with Shifted Vertex

Common Pitfall: Applying $a+b+c=0$ for orthogonal generators when the vertex is not at the origin.

Counterexample: Cone with vertex $(\alpha, \beta, \gamma) \neq (0,0,0)$

Why it fails: The condition $a+b+c=0$ applies strictly to cones whose vertex is the origin. For shifted vertices, origin must be shifted first.

🛡️ Exam Defense: Always check the vertex coordinates given in the question before applying standard coefficient sum rules.

⚠️ COMMON ERROR #02

Assuming Degenerate Cones are Regular

Common Pitfall: Treating $\Delta = 0$ equations as standard valid cones without checking if they split into planes.

Counterexample: $x^2 - y^2 = 0$ in 3D ($x^2 - y^2 + 0z^2 = 0$)

Why it fails: The determinant of the coefficient matrix is zero, meaning the surface represents two intersecting planes, not a proper cone.

🛡️ Exam Defense: Always compute the determinant of the 3x3 matrix of second-degree coefficients; if $\det = 0$, it is a degenerate cone.

Comparison

Concept / Theorem	Hypotheses / Assumptions	Key Formula / Outcome	Geometric Meaning	Exam Identification Clue
General Cone (Origin)	Homogeneous 2nd degree equation	$ax^2+by^2+cz^2+2fyz+2gzx+2hxy=0$	Vertex at $(0,0,0)$	No linear or constant terms
Right Circular Cone	Constant semi-vertical angle $\alpha$	$(lx+my+nz)^2 = (l^2+m^2+n^2)(x^2+y^2+z^2)\cos^2\alpha$	Symmetric about axis line	Mentions axis and semi-vertical angle
Enveloping Cone	Cone touching surface along a curve	$SS_1 = T^2$	Tangent lines drawn from external point	Vertex given, touching a sphere/conicoid
Reciprocal Cone	Reciprocal set of generators	$Ax^2+By^2+Cz^2+2Fyz+2Gzx+2Hxy=0$	Perpendicular tangent planes form generators	Cofactor matrix utilization

Solved examples

Each problem includes a full solution and a shorter method:

CANONICAL PROBLEM 01 Dual-Method Breakdown

Find the condition that the cone $ax^2 + by^2 + cz^2 + 2fyz + 2gzx + 2hxy = 0$ has three mutually perpendicular generators.

(A) $a+b+c = 0$(B) $abc + 2fgh - af^2 - bg^2 - ch^2 = 0$(C) $f+g+h = 0$(D) $ab+bc+ca = 0$

📚 Method A: Formal Textbook Derivation

Let the three mutually perpendicular generators have direction cosines $(l_1, m_1, n_1), (l_2, m_2, n_2), (l_3, m_3, n_3)$. Since they satisfy the cone equation and are orthogonal, summing the equations yields $a+b+c = 0$.

⚡ Method B: 25-Second Exam Shortcut

Direct standard textbook result; recall instantly as Shortcut 1 for 5 seconds execution.

💡 Tip: Memorize $a+b+c=0$ for orthogonal generators of origin-vertex cones.

CANONICAL PROBLEM 02 Dual-Method Breakdown

Find the equation of the right circular cone whose vertex is the origin, axis is the line $\frac{x}{1} = \frac{y}{2} = \frac{z}{2}$, and semi-vertical angle is $30^\circ$.

(A) $8x^2 + 5y^2 + 5z^2 - 16yz - 8zx - 4xy = 0$(B) $x^2 + y^2 + z^2 - xy - yz - zx = 0$(C) $3x^2 + 4y^2 + 4z^2 = 0$(D) $5x^2 + 8y^2 + 8z^2 = 0$

📚 Method A: Formal Textbook Derivation

Step 1: Identify axis direction ratios $(1, 2, 2)$ and $\cos 30^\circ = \frac{\sqrt{3}}{2}$. Step 2: Apply formula $(lx+my+nz)^2 = (l^2+m^2+n^2)(x^2+y^2+z^2)\cos^2\alpha$. Step 3: Expand and simplify.

⚡ Method B: 25-Second Exam Shortcut

Substitute point $(1, 0, 0)$ or test homogeneity and axis symmetry to eliminate options.

💡 Tip: Check $\cos^2 30^\circ = \frac{3}{4}$ multiplier on the magnitude term.

Practice questions

Choose an answer to check it and see the solution:

The equation $2x^2 - 3y^2 + z^2 = 0$ represents:

A A cone with vertex at origin

B A cylinder

C A pair of planes

D A sphere

Derivation & Analysis:

It is a homogeneous equation of second degree in three variables with no linear terms, representing a cone with vertex at the origin.

If $ax^2 + by^2 + cz^2 + 2fyz + 2gzx + 2hxy = 0$ has mutually perpendicular generators, then:

A $abc = 0$

B $a+b+c = 0$

C $f=g=h=0$

D $ab+bc+ca = 0$

Derivation & Analysis:

The standard condition for three mutually perpendicular generators of a homogeneous cone is $a+b+c=0$.

The semi-vertical angle of a right circular cone whose axis is the z-axis is given by $\tan \alpha = \frac{\sqrt{x^2+y^2}}{z}$. For $\alpha = 45^\circ$, the equation is:

A $x^2 + y^2 - z^2 = 0$

B $x^2 + y^2 + z^2 = 0$

C $x^2 - y^2 - z^2 = 0$

D $2x^2 + y^2 = z^2$

Derivation & Analysis:

When $\alpha = 45^\circ$, $\tan 45^\circ = 1$, so $\sqrt{x^2+y^2} = z$, leading to $x^2 + y^2 - z^2 = 0$.

The enveloping cone of the sphere $x^2 + y^2 + z^2 = a^2$ with vertex at $(x_1, y_1, z_1)$ is given by:

A $(x^2+y^2+z^2-a^2)(x_1^2+y_1^2+z_1^2-a^2) = (xx_1+yy_1+zz_1-a^2)^2$

B $SS_1 = T^2$

C $xx_1 + yy_1 + zz_1 = a^2$

D $x^2+y^2+z^2 = a^2$

Derivation & Analysis:

Symbolically, the enveloping cone equation is always given by $SS_1 = T^2$.

The plane $lx + my + nz = 0$ cuts the cone $ax^2 + by^2 + cz^2 = 0$ in perpendicular generators if:

A $a(b+c)l^2 + b(c+a)m^2 + c(a+b)n^2 = 0$

B $al^2 + bm^2 + cn^2 = 0$

C $l^2 + m^2 + n^2 = 0$

D $al + bm + cn = 0$

Derivation & Analysis:

Eliminating $z$ between the plane and cone and applying the perpendicularity condition $\text{coeff}(x^2) + \text{coeff}(y^2) = 0$ yields $a(b+c)l^2 + b(c+a)m^2 + c(a+b)n^2 = 0$.

A cone is a ruled surface generated by a line which passes through a fixed point called:

A Directrix

B Vertex

C Axis

D Generator

Derivation & Analysis:

By definition, the fixed point through which all generating lines pass is the vertex of the cone.

If the equation $ax^2 + by^2 + cz^2 + 2fyz + 2gzx + 2hxy = 0$ has discriminant $\Delta = 0$ (where $\Delta = abc + 2fgh - af^2 - bg^2 - ch^2$), the cone represents:

A A real right circular cone

B An imaginary cone

C A pair of intersecting planes

D A sphere

Derivation & Analysis:

When the determinant of the quadratic form vanishes ($\Delta = 0$), the cone degenerates into a pair of intersecting planes.

The reciprocal cone of $ax^2 + by^2 + cz^2 = 0$ is:

A $ax^2 + by^2 + cz^2 = 0$

B $bcx^2 + cay^2 + abz^2 = 0$

C $a^2x^2 + b^2y^2 + c^2z^2 = 0$

D $x^2/a + y^2/b + z^2/c = 0$

Derivation & Analysis:

The cofactors of $a, b, c$ in $ax^2+by^2+cz^2=0$ are $bc, ca, ab$, giving the reciprocal cone $bcx^2 + cay^2 + abz^2 = 0$.

The general equation of second degree representing a cone with vertex at $(\alpha, \beta, \gamma)$ is obtained by shifting the origin. The degree of the homogeneous terms in shifted coordinates is:

A 1

B 2

C 3

D 0

Derivation & Analysis:

After shifting the origin to the vertex, the equation becomes strictly homogeneous of degree 2 in the new variables.

Q10

What is the angle between the lines in which the plane $ux + vy + wz = 0$ cuts the cone $ax^2 + by^2 + cz^2 = 0$?

A $\tan \theta = \frac{2\sqrt{a b c (...)}}{...}$

B Depends on principal axes

C Given by $\tan \theta = \frac{2\sqrt{(bu^2+cv^2+aw^2)(...)}}{...}$

D Always $90^\circ$

Derivation & Analysis:

The standard intersection angle formula involves the discriminant terms of the planar section with coefficients $bu^2 + cv^2 + aw^2$, etc.

Common questions

A general cone is represented by a homogeneous second-degree equation $ax^2 + by^2 + cz^2 + 2fyz + 2gzx + 2hxy = 0$ without any specified symmetry about a single axis. A right circular cone specifically requires an axis line and a constant semi-vertical angle $\alpha$.

Candidates often apply $a+b+c=0$ blindly when the vertex is shifted away from the origin $(0,0,0)$. The condition $a+b+c=0$ is valid strictly when the vertex is at the origin.

Compute the determinant $\Delta = abc + 2fgh - af^2 - bg^2 - ch^2$. If $\Delta \neq 0$, it is a proper cone; if $\Delta = 0$, it represents a pair of intersecting planes.

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Cylinder