Vectors and Geometry Part 19

Vectors and geometry part 19

presented and prepared by Rashmi-m

content edited by nandakumar

The conicoids

Objectives

1 introduces the idea of conicoids

2.studies the nature and shape of some of the surfaces

Introduction

In this session we introduce the concept of conicoids or say,quadric surface which is the path of the general equation of second degree .then we move on to the transformation of second degree equation to different standard forms like ellipsoid ,hyperboloid , cone,etc.in the coming sessions we discuss more about conicoids.

Definition: The locus of a general equation of second

degree

ax2 +by2+cz2+2fyz+2gzx+2hxy+2ux+2vy+2wz+d=0 ……..(1)

is called a quadric surface or a conicoid.

The general equation of the second degree contains ten constants, but only nine are effective .thus a conicoid can be determined to satisfy nine conditions each of which give rise to one relation between the constants but it is not easy to determine the characteristics and location of a quadric surface corresponding to the general equation (1).therefore a suitable transformation of the coordinate axes is adopted to reduce the general equation of the second degree to a number of standard forms like Ellipsoid, Hyperboloid of one sheet ,Hyperboloid of two sheet ,etc Now let us discuss the nature and shape of the surfaces.

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The Ellipsoid.

The locus of the equation

……………..(2)

Is called an ellipsoid

(see fig 1)

The centre; If be the coordinates of any point on (2), then

This shows that the point also lies on (2). But these points are on a straight line through the origin and are equidistant from the origin (as origin is the middle point of the line joining these points. Hence the origin bisects every chord which passes through it and is called the centre of the surface. Thus the origin is the centre of ellipsoid (2).

(ii) SymmetryIf the point with coordinates lies on (2), then so does also the point. The middle point of the line joining these points is. This point lies on the XOY plane (i.e. plane) and also the line is perpendicular to this plane. Hence the XOY plane bisects every chord perpendicular to it and therefore the surface is symmetrical with respect to this plane.

Similarly, it can be shown that the surface is symmetrical with respect to the YOZ and the ZOX planes.

These three planes are called the principal planes of the ellipsoid. The three lines of intersection of the three principal planes, taken in pairs, are called the principal axes of the ellipsoid. In this case, the coordinate axes are the principal axes.

(iii) Surface is closed. Let in (2) be in the descending order of magnitude, so that being on the surface, we see that

and

This shows that no point on the surface is at a distance greater than a, or less than c from the origin. The surface is therefore limited in every direction. Also the x-coordinate of any point on the surface cannot numerically exceed a, for otherwise the first term alone in (2) would be greater than 1 and thus either y or z will be imaginary. Similarly the y- coordinate and z- coordinates of any point cannot numerically exceed b and c respectively. Therefore the surface is bounded by the planes.

i.e., the surface is closed.

(iv) Section of the surface. The section of the surface (2) by the plane (parallel to XOY plane) is given by the equations.

This is an ellipse with its semi-axes

And centre on the z-axis.

This is a real ellipse if , a point ellipse if and an imaginary

ellipse if .

The surface is therefore generated by a variable ellipse whose plane is parallel to the plane and whose centre lies on z-axis. This ellipse diminishes in size as k varies from 0 to c or from 0 to –c.

Similarly it can be shown that the sections by the planes parallel to other coordinate planes are also ellipses and the ellipsoid is generated by them.

Note. If two of the three quantities a, b and c in the equation of the surface are equal, the sections parallel to one of the coordinate planes are circles. Thus taking and choosing a permissible value for x, we have the equation.

of a circle parallel to the YOZ plane. The equation

of the surface therefore represents an ellipsoid of

revolution formed by revolving the ellipse

about its major axis. This is known as a prolate spheroid.

Note. A surface generated by revolving a plane curve about a straight line is called a surface of revolution.

(v) Lengths of the axes: The x-axis meets the surface in the two points and. Thus the surface intercepts a length 2a on the x-

axis. Similarly the lengths intercepted on y- and z- axes are 2b and 2c respectively. The lengths 2a, 2b, 2c intercepted on the principal axes are called the lengths of the axes of the ellipsoid.

(B) The Hyperboloid of one sheet. The surface represented by the equation

… (3)

is called a hyperboloid of one sheet.

nature and shape of the surface see fig (2)

(i) Origin is the centre of the surface represented by (3)

(ii) The surface is symmetrical with respect to each of the coordinate planes are the principal planes of the surface. The coordinate axes are its principal axes.

(iii) The surface is obviously unbounded, for each of the coordinate x, y and z can take any values provided only that

is not less than unity.

(iv)The section of the surface by the plane (parallel to XOY plane) is given by the equations

The section is , therefore an ellipse whose semi-axes are

and whose centre lies on z-axes and which increases in size as k increases. There is no limit to the increase of k. The surface may, therefore, be generated by the variable ellipse parallel to the XOY plane which increases in size as it moves rather away from the coordinate planes i.e., as k varies from 0 to or from 0 to .

But the sections of the surface parallel to the other coordinate planes are not ellipses. Thus the sections by the planes (parallel to YOZ plane) and

(parallel to ZOX plane) are hyperbolas

and

respectively.

(v) The x-axis meets the surface in the two points and and thus the length intercepted on x-axis is . Similarly, the length intercepted on y-axis is 2b, whereas the z-axis doesnot meet the surface in real points.

© Hyperboloid of two sheets .The surface represented by

…………….(4)

Is called an hyperboloid of two sheets .similarly to the case of ellipsoid we can prove the nature and the shape of the hyperboloid of two sheets see fig(3)

(i)origin is the centre of the surface represented by (4)

(ii)The surface is symmetric with respect to the coordinate plane .the coordinate planes are the principle planes and the coordinate

axes principle axes of the surface

(iii)The surface is unbounded in all the coordinates x,y,z but the intersection with y- axes and z- axes are imaginary

(iv)The section of the surface by the plane x=k is given by the equation

The section is therefore ,an ellipse with semi axes

It is a real ellipse if k2=a2

and an imaginary ellipse if .Thus there is no portion of the surface included between the planes x=-a and x=a. for x2>a2 the section

is a real ellipse and it increase in size as k increases

The section by the planes y=k and z=k are the hyperbolas

respectively.These hyperbolas extent to infinity as k varies from o to infinity or from 0 to .thus the surface may also be considered as generated by

variable hyperbolas parallel to ZOX and XOY planes and is consequently called a hyperboloid of two sheets

(v)The x - axis meets the surface in the two points (a,0,0) and (-a,0,0) and therefore the length intercepted on x-axis is 2a

Whereas the y and z –axes meet the surface in imaginary points .

Example 1 Find the equation of the tangent plane to the conicoid at at the point.

Solution

The equation of the tangent plane to the given conicoid at is

The required tangent plane at is

or.

Example 2 Tangent planes are drawn to the conicoid through. Show that the perpendiculars from the centre of the conicoid to these planes generate the cone

Solution

Any plane through is

or… (5)

If it is a tangent plane to the given conicoid, then

… (6)

Also the equation of the line through the centre (0,0,0) of the conicoid perpendicular to (5) is … (7)

Its locus is obtained by eliminating between (6) and (7) and is

Example 3:Find the equation of the tangent planes to 7x2+5y2+3z2=0 which pass through the line 7x+10y=30,5y-3z=0

Solution :Any plane through the line

7x+10y-30 =0,5y-3z=0

If it touches the conicoid

7x2+5y2+3z2=60

Then by

We get

That is

Which implies

Summary

In this session we discussed the quadric surfaces which are locus of the general second degree .followed by some of the nature and geometric properties of the surfaces like ellipsoid , hyperboloid etc,

Assignment

1.explain the nature and geometric properties of elliptical paraboloid and hyperbolic paraboloid

2.Discuss the shape of the surface imaginary cone

3.explain elliptical cylinder.

Glossary

Conicoid:The locus of a general equation of second degree

ax2 +by2+cz2+2fyz+2gzx+2hxy+2ux+2vy+2wz+d=0

is called a quadric surface or a conicoid.

FAQ

1.In the discussion of ellipsoid what happened when the minor axis is revolved

answer

The equation

represents the ellipsoid formed by revolving the same ellipse about its minor axis and is known as an oblate spheroid.

If all the three quantities a, b and c in the surface are equal i.e., , the ellipsoid becomes a sphere.

QUIZ

1.The standard form of ellipsoid

(a) (b) (c)(d)

2 .The surface represented by the equation

(a)ellipsoid (c)hyperboloid (b)hyperboloid of one sheet (c)cone

3. A surface generated by revolving a plane curve about a

straight line is called

Answer

1 (a)

2 (c)hyperboloid of one sheet

3 (a) surface of revolution

REFERENCE

1S.L.Loney The Elements of Coordinate Geometry ,Macmillian and company, London

2Gorakh Prasad and H.C.Gupta Text book of coordinate geometry, Pothisalapvt ltd Allahabad

3P.K.Mittal Analytic Geometry Vrinda Publication pvtLtd,Delhi.