Conic section

1556 words (6 pages) Essay

1st Jan 1970 Mathematics Reference this

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Conic Section

The names parabola and hyperbola are given by Apolonius. These curve are infact, known as conic sections or more commonly conics because they can be obtained as intersections of a plane with a double napped right circular cone. These curves have a very wide range of applications in fields such as planetary motion, design of telescope and antenna, reflectors in flash light and automobiles headlights etc.

Section of a cone:-

Let l be a fixed vertical line and ‘m’ be another line intersecting it at a fixed point ‘v’ and inclined to it at an angle α.

Suppose we rotate the line ‘m’ around line 1 in such a way that the angle α remains constant. Then the surface generated is a doubly-napped right circular hollow cone.

The point V is called the vertex; the line 1 is the axis of the cone. The rotating line ‘m’ is called a generator of the cone. The vertex separates the cone into two parts called nappes.

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If we take the intersection of a plane with a cone, the section so obtained is called conic section. Thus, conic sections are the curve obtained by intersecting the right circular cone by a plane.

We obtain different type of conic section depending on the position of the intersecting plane with respect to the cone and by the angle made by the intersecting plane with the vertical axis of the cone.

The intersect of the plane with the cone can take place either at vertex or any other part of the nappe either below or above the vertex.

Circle, Ellipse, Parabola, Hyperbola

When the plane cuts the nappe of the cone, we have the following situation –

  1. When β = 90 degree, the section is circle.
  2. α < β < 90 degree, the section is ellipse.
  3. β = α, the section is parabola.

(In each of above three situation, the plane cuts entirely across on enappe of the cone )

d) 0 ≤ β < α ; the plane cuts through both the

Nappe & the curve of intersection is a hyperbola.

Degenerated conic section

When the plane cuts at the vertex of the cone, we have the following different cases.

  1. When α < β ≤ 90 degree, the section is a point.
  2. When β = α, the plane contains the generator of the cone & the section is a straight line. ( It is the degenerated case of parabola )
  3. When 0 ≤ β < α , the section is a pair of intersecting straight lines.

(It is degenerated form of a hyperbola).

Summary of Basic Properties

Circle

Ellipse

Parabola

Hyperbola

Standard Cartesian Equation :

x2 + y2 = r2

y2 = 4ax

Eccentricity (e):

0

0 < e <1

1

1 < e

Relation between a,b

and e

b = a

b2 = a2(1-e2)

b2 = a2(e2-1)

Parametric Representation

x = at2

y = 2at

Or

Definition: It is the locus of all points which meet the condition…

distance to the origin is constant

sum of distances to each focus is constant

distance to focus = distance to directrix

difference between distances to each foci is constant

It might tidy the logic up to consider a circle to be a special case of an ellipse. Then there are two ‘main’ classes

  • an ellipse, with e < 1
  • a hyperbola, with e > 1

And a ‘critical’ class – the parabola with e = 1.

The General Equation of a Conic

The General Equation for a Conic is

Ax2 + Bxy + Cy2 + Dx + Ey + F = 0

The actual type of conic can be found from the sign of B2 – 4AC

If B2 – 4AC is…

Then the curve is a…

< 0

Ellipse, circle, point or no curve

= 0

Parabola, 2 parallel lines, 1 line or no curve.

> 0

> 0

Polar Form

For an origin at a focus, the general polar form (apart from a circle) is

where L is the semi latus rectum.

Ellipse

The Cartesian equation of an ellipse is

where a and b would give the lengths of the semi-major and semi-minor axes.

In its general form, with the origin at the center of coordinates

  • the foci are at
  • the directrix are at
  • the major axis of length 2a
  • the minor axis is of length 2b
  • the semi latus rectum is of length

From the general polar form, the equation for an ellipse is

For any point P on the perimeter, the sum

PF1 + PF2

will be constant, no matter which point is chosen as P.

Hence, an ellipse can also be defined as the locus of a point which moves in a plane so that the sum of its distances from two fixed points is constant. According to Kepler’s First law, the orbit of a planet is an ellipse. The Earth is shaped like an ellipsoid.

Any signal from one of the foci will pass thru the other focus.

Hyperbola

The Cartesian equation of an hyperbola is

In its general form, with the origin at the center of coordinates

  • the foci are at (+/- ae, 0)
  • the directrix are at x = +/- a/e
  • the transverse axis of length 2a
  • the conjugate axis is of length 2b
  • the semi latus rectum is of length 2b2/a

Note the similarity in notation with ellipses; although now the eccentricity is greater than one

Also by analogy with an ellipse

For any point P on a hyperbola, the sum

PF1 – PF2

will be constant, no matter which point is chosen as P.

Hence, a hyperbola can also be defined as the locus of a point which moves in a plane so that the difference of its distances from two fixed points is constant.

Asymptotes of Hyperbola

Rejigging the hyperbola formula to

As x becomes larger, y tends to

These are the equations of the asymptotes.

Similar & Diagonalizable Matrices

Let A and B be square matrices of the same order .The matrix A is said to be similar to the matrix B if their exists an invertible matrix P such that

A=P-1BP or PA=BP ——– (1)

Post multiplying eqn(1) by p-1 , we get

PAP-1 =B

Therefore A is similar to B if and only if B is similar to A. The matrix P is called similarity matrix.

Diagonalizable matrices

A matrix A is diagonalizable, if it is similar to diagonal matrix, that is there exist an invertible matrix P such that P-1AP =D, where D is a diagonal matrix . Since similar matrix has the same eigen values, the diagonal element of D are the eigenvalues of A. A necessary and sufficient condition for the existence of P is given by the following theorem.

Theorem: A square matrix of order n is diagonalizable if and only if it has n linearly independent eigenvectors.

Quadratic Forms –

Let x = (x1 , x2 , ……….xn) be an arbitary vector in IRn. a real quadratic form is an homogeneous expression of the form

Q = ∑ni=1 ∑nj=1 aijxixj

In which the total power in each term is 2. Expanding, we can write

Q = a11x2 + ( a12 + a21 )x1x2 + ……….+ ( a1n +an2 )x2xn +….+annx2n

= xTAx

Using the definition of matrix multiplication. Now, set bij = (aij + aji)/2.

The matrix B = (bij) is symmetric since bij =bji. Further, bij+bji = aij + aji.

Hence,

Q = xTBx

Where B is a symmetric matrix and bij = (aij + aji)/2.

For example, for n = 2, we have

B11 =a11, b12 = (a12 + a21)/2 and b22 = a22.

Ques. — Find out what type of conic section the following quadratic form represents and transform it to principal axes

Q=17×12-30x1x2+17×22=128

We have Q = xT Ax, where,

This gives the characteristic equation (17- λ)2 – 152 = 0.

It has the root λ1= 2, λ2= 32.

Hence it becomes,

Q = 2y12 + 32y22

We see that Q = 128, represents the ellipse.

2y12 + 32y22 = 128, i.e.

y12/82 + y22/22 = 1

If we want to know the direction of the principal axes in the x1x2- coordinate, we have to determine normalized eigen vector from (A- λI)X=0 with

λ = λ1= 2, & λ = λ2 = 32 & then, we get,

&

Hence,

X=Xy =

X1 = y1/2 – y2/2

X2= y1/2 + y2/2

This is 450 rotation.

Conic Section

The names parabola and hyperbola are given by Apolonius. These curve are infact, known as conic sections or more commonly conics because they can be obtained as intersections of a plane with a double napped right circular cone. These curves have a very wide range of applications in fields such as planetary motion, design of telescope and antenna, reflectors in flash light and automobiles headlights etc.

Section of a cone:-

Let l be a fixed vertical line and ‘m’ be another line intersecting it at a fixed point ‘v’ and inclined to it at an angle α.

Suppose we rotate the line ‘m’ around line 1 in such a way that the angle α remains constant. Then the surface generated is a doubly-napped right circular hollow cone.

The point V is called the vertex; the line 1 is the axis of the cone. The rotating line ‘m’ is called a generator of the cone. The vertex separates the cone into two parts called nappes.

If we take the intersection of a plane with a cone, the section so obtained is called conic section. Thus, conic sections are the curve obtained by intersecting the right circular cone by a plane.

We obtain different type of conic section depending on the position of the intersecting plane with respect to the cone and by the angle made by the intersecting plane with the vertical axis of the cone.

The intersect of the plane with the cone can take place either at vertex or any other part of the nappe either below or above the vertex.

Circle, Ellipse, Parabola, Hyperbola

When the plane cuts the nappe of the cone, we have the following situation –

  1. When β = 90 degree, the section is circle.
  2. α < β < 90 degree, the section is ellipse.
  3. β = α, the section is parabola.

(In each of above three situation, the plane cuts entirely across on enappe of the cone )

d) 0 ≤ β < α ; the plane cuts through both the

Nappe & the curve of intersection is a hyperbola.

Degenerated conic section

When the plane cuts at the vertex of the cone, we have the following different cases.

  1. When α < β ≤ 90 degree, the section is a point.
  2. When β = α, the plane contains the generator of the cone & the section is a straight line. ( It is the degenerated case of parabola )
  3. When 0 ≤ β < α , the section is a pair of intersecting straight lines.

(It is degenerated form of a hyperbola).

Summary of Basic Properties

Circle

Ellipse

Parabola

Hyperbola

Standard Cartesian Equation :

x2 + y2 = r2

y2 = 4ax

Eccentricity (e):

0

0 < e <1

1

1 < e

Relation between a,b

and e

b = a

b2 = a2(1-e2)

b2 = a2(e2-1)

Parametric Representation

x = at2

y = 2at

Or

Definition: It is the locus of all points which meet the condition…

distance to the origin is constant

sum of distances to each focus is constant

distance to focus = distance to directrix

difference between distances to each foci is constant

It might tidy the logic up to consider a circle to be a special case of an ellipse. Then there are two ‘main’ classes

  • an ellipse, with e < 1
  • a hyperbola, with e > 1

And a ‘critical’ class – the parabola with e = 1.

The General Equation of a Conic

The General Equation for a Conic is

Ax2 + Bxy + Cy2 + Dx + Ey + F = 0

The actual type of conic can be found from the sign of B2 – 4AC

If B2 – 4AC is…

Then the curve is a…

< 0

Ellipse, circle, point or no curve

= 0

Parabola, 2 parallel lines, 1 line or no curve.

> 0

> 0

Polar Form

For an origin at a focus, the general polar form (apart from a circle) is

where L is the semi latus rectum.

Ellipse

The Cartesian equation of an ellipse is

where a and b would give the lengths of the semi-major and semi-minor axes.

In its general form, with the origin at the center of coordinates

  • the foci are at
  • the directrix are at
  • the major axis of length 2a
  • the minor axis is of length 2b
  • the semi latus rectum is of length

From the general polar form, the equation for an ellipse is

For any point P on the perimeter, the sum

PF1 + PF2

will be constant, no matter which point is chosen as P.

Hence, an ellipse can also be defined as the locus of a point which moves in a plane so that the sum of its distances from two fixed points is constant. According to Kepler’s First law, the orbit of a planet is an ellipse. The Earth is shaped like an ellipsoid.

Any signal from one of the foci will pass thru the other focus.

Hyperbola

The Cartesian equation of an hyperbola is

In its general form, with the origin at the center of coordinates

  • the foci are at (+/- ae, 0)
  • the directrix are at x = +/- a/e
  • the transverse axis of length 2a
  • the conjugate axis is of length 2b
  • the semi latus rectum is of length 2b2/a

Note the similarity in notation with ellipses; although now the eccentricity is greater than one

Also by analogy with an ellipse

For any point P on a hyperbola, the sum

PF1 – PF2

will be constant, no matter which point is chosen as P.

Hence, a hyperbola can also be defined as the locus of a point which moves in a plane so that the difference of its distances from two fixed points is constant.

Asymptotes of Hyperbola

Rejigging the hyperbola formula to

As x becomes larger, y tends to

These are the equations of the asymptotes.

Similar & Diagonalizable Matrices

Let A and B be square matrices of the same order .The matrix A is said to be similar to the matrix B if their exists an invertible matrix P such that

A=P-1BP or PA=BP ——– (1)

Post multiplying eqn(1) by p-1 , we get

PAP-1 =B

Therefore A is similar to B if and only if B is similar to A. The matrix P is called similarity matrix.

Diagonalizable matrices

A matrix A is diagonalizable, if it is similar to diagonal matrix, that is there exist an invertible matrix P such that P-1AP =D, where D is a diagonal matrix . Since similar matrix has the same eigen values, the diagonal element of D are the eigenvalues of A. A necessary and sufficient condition for the existence of P is given by the following theorem.

Theorem: A square matrix of order n is diagonalizable if and only if it has n linearly independent eigenvectors.

Quadratic Forms –

Let x = (x1 , x2 , ……….xn) be an arbitary vector in IRn. a real quadratic form is an homogeneous expression of the form

Q = ∑ni=1 ∑nj=1 aijxixj

In which the total power in each term is 2. Expanding, we can write

Q = a11x2 + ( a12 + a21 )x1x2 + ……….+ ( a1n +an2 )x2xn +….+annx2n

= xTAx

Using the definition of matrix multiplication. Now, set bij = (aij + aji)/2.

The matrix B = (bij) is symmetric since bij =bji. Further, bij+bji = aij + aji.

Hence,

Q = xTBx

Where B is a symmetric matrix and bij = (aij + aji)/2.

For example, for n = 2, we have

B11 =a11, b12 = (a12 + a21)/2 and b22 = a22.

Ques. — Find out what type of conic section the following quadratic form represents and transform it to principal axes

Q=17×12-30x1x2+17×22=128

We have Q = xT Ax, where,

This gives the characteristic equation (17- λ)2 – 152 = 0.

It has the root λ1= 2, λ2= 32.

Hence it becomes,

Q = 2y12 + 32y22

We see that Q = 128, represents the ellipse.

2y12 + 32y22 = 128, i.e.

y12/82 + y22/22 = 1

If we want to know the direction of the principal axes in the x1x2- coordinate, we have to determine normalized eigen vector from (A- λI)X=0 with

λ = λ1= 2, & λ = λ2 = 32 & then, we get,

&

Hence,

X=Xy =

X1 = y1/2 – y2/2

X2= y1/2 + y2/2

This is 450 rotation.

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