2. 2
ConeUnit V
- Right Circular Cone
- Intersection of a Straight line and a Quadric cone
- Tangent Plane and Normal
- Condition for the plane to touch the Quadric cone
3. Cone
Definition :
A surface generated by a straight line passing through a
fixed point and intersecting a given curve is called as a cone.
The fixed point is called the vertex of the cone .
The given curve is called the guiding curve.
A line which generates the cone is called a generator.
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V
A B
CD
The surface in the Fig. 9.1 is a cone with vertex
V, The line V A as a generator. The lines V B, V
C are also generators. In fact every line joining
V to any point of the guiding curve is a
generator of the cone.
4. CONE
Remark :
If the guiding curve is a plane curve of degree n, then the
equation of the cone is also of degree n and we call it a cone of
order n.
Remark:
The equation of a quadratic cone is of second degree in
x, y and z; i.e.,
ax2 +by2 +cz2 +2fyz +2gzx+ 2hxy +2ux+ 2vy +2wz +d = 0.
Theorem :
The equation of a cone with vertex at the origin is a
homogeneous second degree equation.
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5. CONE
Theorem:
Every second degree homogeneous equation in x, y, z
represents a cone with vertex at the origin.
Remark :
If the line with equation, (x/l) = (y/m) = (z/n) is a generator
of a cone with vertex at the origin, then the direction ratios l, m, n
satisfies the equation of the cone.
Remark :
The general equation of a quadratic cone with vertex at the
origin is,
ax2 + by2 + cz2 + 2fyz + 2gzx + 2hxy = 0.
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6. The Right Circular Cone
Definition :
A right circular cone is a surface generated by a
straight line passing through a fixed point and making
a constant angle θ with a fixed straight line passing
through the given point.
The fixed point is called as the vertex of the right
circular cone
The fixed straight line is called as the axis of the cone.
The constant angle θ is called as the semi − vertical
angle of the cone.
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7. Remark :
Every section of a right circular cone by a plane perpendicular
to its axis is a circle.
Let θ be the semi - vertical angle of the cone and α be a plane
perpen- dicular to the axis V N ( see Fig) of the cone. We show
that the section of the cone by the plane α is a circle.
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α
AN
B
V Let P be any point of the section of the cone by the plane
α. Let A be the point of intersection of the axis V N and the
plane α. Then AP s perpendicular to V A. In the right
angled triangle V AP , tan θ = AP/AV.
∴ AP = AV tan θ. Since, AV and tan θ are constant. it
follows that AP is constant for all points P on the section
of the cone by the plane α. Thus the section is a circle.
