Formulae,
Cone,
Right-Circular Cone,
Cylinder,
Right-Circular Cylinder,
The Sphere.

**THREE
DIMENSIONAL ANALYTICAL GEOMETRY **

__Formulae__

__Cone__

__Right-Circular Cone__

__Cylinder__

__Right-Circular Cylinder__

__The Sphere__

**FORMULAE:
**

1. The equation of the straight line
through the point *p*(x_{1},y_{2},z_{1})
and having direction cosines

2. The equation of the straight line
through the point B(x_{2},y_{2},z_{2}) and having
direction ratios

3. The equation of the straight line
passing through the points A(x_{1},y_{1},z_{1}) and B(x_{2},y_{2},z_{2})
is

4. Angle between the straight lines:

**CONE**

**DEFINITION:**
A cone is defined as a surface generated by a straight line which passes
through a fixed point and satisfies one or more conditioni.e.ie, it may
intersect a fixed curve.

Note:

1. The fixed point is said to be the
vertex of the cone

2. The fixed curve is said to be the
guiding curve of the cone

3. The straight line in any position is
called the generator of the cone.

**FORMULA:**

The equation of the cone with vertex (x_{1},y_{2},z_{1})
and whose generators intersect the guiding curve

**1.
Find the equation of the cone with vertex at (1,1,1) and which passes through
the curve given by **

**RIGHT-CIRCULAR CONE**

**DEFINITION:**
A right circular cone is a surface generated by a straight line which passes
through a fixed point and makes a constant angle with a fixed line through the
fixed point. The equation of right
circular cone vertex is (x_{1},y_{1},z_{1}) ,the semi
vertical angle a and axis the line

**CYLINDER**

**DEFINITION:**
A cylinder is a surface generated by a straight line which is parallel to a
fixed line and it has to intersect a given fixed curve. The straight line is
any position called a generator and the fixed point is called the guiding curve
of the cylinder.

The equation of cylinders whose generators are parallel to the line

**RIGHT CIRCULAR
CYLINDER**

DEINITION: Right circular cylinder is a
surace generated by a straight line which is parallel to a fixed line is at a
contant distance it or whose guiding
curve is a circle.

**THE
SPHERE**

**DEFINITION:**
A sphere is the locus of a point moving at a constant distance form a fixed
point. The constant distance is the
radius and the fixed point is the centre of the sphere.

**PLANE
SECTION OF A SPHERE: **

A
plane section of a sphere is a circle sphere S: x^{2}+y^{2}+z^{2}+2ux+2vy+2wz+d=0
plane U: ax+by+cz+d_{1}= 0 the combined equation (S,U) is a circle.

The equation of the sphere through the circle
a (S, U ) is S_{1}=S+KU

**EQUATION
OF THE TANGENT PLANE **

The sphere is x^{2}+y^{2}+z^{2}+2ux+2vy+2wz+d=0
and the point of contact is (x_{1},y_{1},z_{1})
then
Equation of the Tangent plane is xx_{1}+yy_{1}+zz_{1}+
u(x+x_{1})+v(y+y_{1})+w(z+z_{1}) +d=0

**CONDITION
FOR TANGENCY: **

Condition for tangency is perpendicular from
centre to the plane = radius

**CONDITION
FOR ORTHOGONALITY OF TWO SPHERES: **

The condition for orthogonality of two spheres

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