Chapter 1-Conics

**Conic section**- Shapes that are obtained by taking different plane slices through a __double cone__

__Menaechmus__ (Greek mathematician) discovered conic sections around __350 BC__

Slice 7 is considered an ellipse

**Ellipse-**An area where the sum of the distances from 2 fixed points on the plane remain constant

The

__ellipse__and the__hyperbola__both have a**centre****Non-degenerate conics**- conics that are__parabolas, ellipses, or hyperbolas__**Degenerate**- A single point, line, and pair of lines

**Circle**- a__type of conic,__a set of points that lie at a fixed distance (radius), from a fixed point (centre)Use the techniques of coordinate geometry to find the

__equation of a circle__with given centre and radius.**Theorem 1**: The equation of a circle with centre (a,b) and radius r is (x-a)^2 + (y-b)^2 = r^2Why?

This is a variation of the distance formula (d= (x1-x2)^2 + (y1-y2)^2)

Write the equation for the given centre and radius

(a) centre the origin, radius 1

(1)^2 - (x-0)^2 + (y-0)^2

1= x^2 + y^2

**x^2 + y^2 -1 = 0**

(b) centre (3, 4), radius 3

(3)^2 = (x-3)^2 + (y-4)^2

9= (x-3)(x-3)+ (y-4)(y-4)

9 = x^2 + y^2 -6x -8y + 25

**x^2 + y^2 -6x -8y +16=0**

Determine the condition on the numbers f, g, and h in the equation x^2 + y^2 + fx + gy + h = 0 for the equation to pass through the __origin__

This is the

__factored/ distributed form__of our original equationWe have to find the state of the equation when

**a and b equal zero**…. (x-0)^2 and (y-0)^2We can factor our this form to see the conditions

This would mean that

**f,g, and h all must equal 0**

(c) Consider the set of points (x,y) that satisfy the equation: x^2 + y^2 -4x +6y +9 = 0

Our goal is to complete the square (note that the coefficients on x^2 and y^2 are both 1)

Re-write equations containing just the x-terms, and just the y-terms

x^2 − 4x = (x − 2)^2− 4

y^2 + 6y = (y + 3)^2 − 9

These terms can be substituted into our original equation

(x-2)^2 -4 + (y+3)^2 -9 +9 = 0

(x-2)^2 + (y+3)^2 = 4

This means that the circle has a radius of 2 and a centre of (2, -3)

**Theorem 2**: You can __complete the square__ in the equation x^2 + y^2 + fx + gy + h = 0 to put it in the form (x+1/2f)^2 + (y+1/2g)^2 = 1/4 f^2 +1/4g^2 -h

This equation represents a circle with the centre (-1/2f, -1/2g) and a radius sqrt(1/4f^2 + 1/4g^2 -h)

This only occurs when the 1/4f^2 + 1/4g^2 -h is greater than zero, if this is less than zero, we can prove that the figure is __not a circle__

**Why?**

The equation is equal to r^2 (radius squared), when any number is squared, it has to equal a positive number. Also, when you go to take the square root of this side of the equation, it is impossible to get a rational answer when square-rooting a negative number.

(d) Determine the centre and radius of the equation

3x^2 + 3y^2 − 12x − 48y = 0.

(e) Determine the set of points that satisfies the following equation

x^2 + y^2 − 2x + 4y + 5 = 0;

Two intersecting circles that meet at __right angles__

**Theorem 3**: __Orthogonality__ test

Two intersecting circles with equations x^2 + y^2 + f1x + g1 y + h1 = 0 and x^2 + y^2 + f2x + g2 y + h2 = 0, are only orthogonal if f1 f2 + g1g2 = 2(h1 + h2).

Why?

This can be tested with the __pythagorem theorem__, where P is a point of intersection, and A and B are points inside of each of the circles. This means that

AP^2 + BP^2 = AB^2. Using our previous formulas, we know that

This can be solved to result in the original equation

In this image, the circle intersects at distinct points P and Q

This equation represents a circle when k __does not__ equal -1, and this means that P and Q will satisfy the equation

These circles have distinct points of intersection, and the line containing these points can be expressed using the second equation, only when __k=-1.__ When k is any other number, this is the equation of __one of the circles__

The three

__non-degenerate conics__are the parabola, ellipse, and hyperbolaThe distance of a set of points from a fixed point (

**focus**) follow a constant multiple (**eccentricity)**This distance is also followed from a line (

**directrix**)

**Eccentricity**- It is an ellipse if e (eccentricity) is between 0 and 1, a parabola of e equals 1, and a hyperbola is e is greater than 1

The set of points (P) whose distance from a fixed point (F) is equal to their distance from a fixed line (D)

Standard form:

The focus must lie on the x-axis, and have coordinates

**(a,0)**and a must be greater than 0The directrix must be

**x=-a**

y^2=4ax

This equation is used to show that each

__real number__corresponds with__one number__on the parabola

The focus and directrix can be described by

x = at^2, y = 2at (t ∈ R).

The x-axis is the axis of the parabola in standard form because the parabola is

__symmetric__in respect to this line.The

__origin__is the__vertex__of the parabola, and it has no centre

e is greater than or equal to zero, but less than one

A set of points whose distance from a certain point is e times their distance from a fixed line

Standard form:

The focus (point) must lie on the x-axis, and have coordinates

**(ae, 0)**, with a being greater than 0The directrix must follow the equation

**x=a/e**

This equation continues to grow, however, as the figure is

__symmetrical.__There is a second focus

**(-ae, 0)**and a second directrix**(-a/e)**The ellipse intersects the axes at the points

**(±a, 0) and (0, ±b).****Major axis**- the segment joining the points (±a, 0)**Minor axis-**the segment joining the points (0, ±b)The minor axis will always be shorter than the major axis because a is greater than b

Parametric equations:

x = a cost, y = b sin t (t ∈ (−π, π]).

e is greater than 1

Standard form:

The focus (point) must lie on the x-axis, and have coordinates

**(ae, 0),**with a being greater than 0The directrix must follow the equation

**x=a/e**

When: b^2 = a^2 (e^2 − 1), a > 0, e > 1.

Foci

**(±ae, 0)**and Directrices**x = ±a/e**Parametric equations: x = a sec t, y = b tan t (t ∈ (−π/2, π/2) ∪ (π/2, 3π/2)).

**Asymptotes-**the lines of the parabola that the branches grow closer to. They are notated as y = ±(b/a)x

e= sqrt (2)

This means that e^2=2 and a=b

The

__asymptotes__have the equation**y = ±x,**so they are at__right angles__A hyperbola whose asymptotes are at right angles is called a

__rectangular hyperbola__

You can use the asymptotes as new

__x and y axes,__and the equation can then be written as xy = c^2The origin will always be at the

__center__in this form

Parametric equations: x = ct, y = c/t where t = 0.

You can describe the equation of a non-degenerate conic in terms of

**polar coordinates**Polar coordinates: r and θ

This formula comes from the equation **OP = e · PM** where the distance from the origin to a point is equal to the eccentricity multiplied by the distance between that same point and the directrix.

**Theorem 5:** The sum of the two focal distances from a point always equal double the major axis.

Why?

The distance from a point to __either foci__ is equal to the __eccentricity times the distance from said point__ __to the directrix.__ This distance is equal to the distance from the second foci to to second directrix **(d’ and f’).** When adding these two distances, they will always equal **2a** (double the major axis)

**Theorem 6:** Differences of __Focal Distances__ of Hyperbola

If a point (P) is closer to

**F**, then**PF’ - PF = 2a**If P is closer to

**F’**then**PF’ - PF = -2a**|PF − PF| is

__constant__for every point on the parabola

A slant plane (pi) that cuts one portion of a right, circular cone is an __ellipse__

Goals: Find the slope of the line that is __tangent__ to the curve at the __parameter__, so that you can determine the __equation of the tangent__

**Theorem 1:** The slope of a tangent to a curve is **y’(t)/ x’(t)**

**Theorem 2:** Equations of tangents in __standard form__

**Polar** of (a,b) with respect to the unit circle

If (a,b) is some point on a tangent, then it can be determined that

**ax1 + by1 = 1**because of the equation**xx1 + yy1 = 1**. This equation can be re-written with points x2 and y2, which deduces the fact that points( x1, y1) and (x2, y2) are on the

__same line,__and this line is called the__polar of (a,b) with respect to the unit circle__

**Theorem 3:** When a point (a,b) lies __outside__ of the circle, and the tangents from point (a,b) touch the circle at P1 and P2, then the equation of the line is **ax + by = 1**

**Normal of a curve**- the line through a point that is__perpindicular to the tangent__of a curve at said pointTo find the equation of a normal of the curve, you find its

__slope__and the__coordinates__of the point on the parabola where they intersect

**Reflection law**- The angle that __incoming light__ makes with the __tangent__ to a surface is the same as the angle that the __reflected light__ makes with the tangent.

Applies to all mirrors, whether

__plane__or__curved__Mirrors are often designed to have a

__conic curve__

**Reflection property of the Ellipse** - light which comes from one focus of an __elliptical mirror__ is reflected at the ellipse to pass through the __second focus.__

**Reflection property of the hyperbola-** light coming from one focus of a hyperbolic mirror is reflected at the __hyperbola__ and makes the light appear to have come from the other focus (**Internal Reflection property**). Light going towards one focus is reflected towards the __other focus__ **(External Reflection Property**)

**Reflection Property of the Parabola**- incoming light __parallel__ to the axis is reflected at the __parabola__ to pass through the __focus__. Light coming from the focus of a parabola is reflected to give a beam of light parallel to the axis of the parabola

__Non-degenerate conics__are the__envelopes__of a family of lines that are tangent to the conicsThe conic being constructed is the

__curve in the plane__that has each of the lines in the family as a tangent

**Auxiliary circle**- A circle whose diameter is its major axisThe tangent to a parabola at its vertex is the auxiliary circle of the parabola

**Theorem 4:**A perpindicular from a focus of a non-degenerate conic to a tangent meets the tangent on the auxiliary circle of the conicWhen given a parabola and its axis, this theorem is used to identify the focus

**Parabola:**

**Ellipse:**

**Hyperbola:**

**Standard form of conics**- the centre is at the __origin__, and the axes are parallel to the __x and y axis__

Equations of all

__non-degenerate conics__follow this equation in__standard form:__Ax^2 + Bxy + Cy^2 + Fx + Gy + H = 0,

Any non-degenerate conic can be determined from a conic in standard form using a

__rotation__**(x, y) → (x cos θ − y sin θ, x sin θ + y cos θ)**followed by a__translation__**(x,y) → (x-a, y-b)**Both tranformations are

__linear__, so the equation of the conic at each stage is just a__second degree__of the original equation

Matrix form of the equation: **x^T Ax + J^T x + H = 0.**

Look back at the old equation: Ax^2 + Bxy + Cy^2 + Fx + Gy + H = 0

The values from this form are used to plug into the

__matrix form__

**Theorem 2:** A 2x2 matrix P represents a __rotation__ of R^2 when P is __orthogonal__ and the det of P = 1

Classify a conic (E) with the equation Ax^2 + Bxy + Cy^2 + Fx + Gy + H = 0

Write the equation in

__matrix__form (**x^T Ax + J^T x + H = 0**)Determine the

__orthogonal matrix__P, with a determinant of**1**that__diagonalizes__AMake the change of coordinate system

**x=Px’**. This changes the equation to**λ1x’^2 + λ2 y’^2 + f x’ + gy’ + h = 0**λ1 and λ2 are the

__eigenvalues__of A

Complete the square to rewrite the equation of E into

**(x’’, y’’)**as the equation of a conic in standard formUse the equation x’ = P^Tx to determine the

__centre__and__axes__of E in the terms of the__original__coordinate system.

**Quadric surfaces**- (quadrics) surfaces in R^3 that are the__natural analogues__of the conics.Ax^2 + By^2 + Cz^2 + Fxy + Gyz + Hxz + Jx + Ky + Lz + M = 0

**Degenerate quadrics**- Empty set, single point, single line, single plane, pair of planes, and a cylinder**Cylinder**- any surface that consists of an ellipse, parabola, or hyperbola in some plane

Use the

__orthogonal diagonalization__of 3x3 matrices to classify non-degenerate quadrics, similarly to how 2x2 matrices were used to__classify conics__

**Idetifying matrices:** In the equation Ax^2 + By^2 + Cz^2 + Fxy + Gyz + Hxz + Jx + Ky + Lz + M = 0,

If a 3x3 matrix (P) represents a __rotation__ of R^3 about the origin when __P is orthogonal__ and **det P = 1**

Classify quadric E with the equation Ax^2 + By^2 + Cz^2 + Fxy + Gyz + Hxz + Jx + Ky + Lz + M = 0,

Write the equation in matrix form (

**x^T Ax + J^T x + M = 0**)Determine the

__orthogonal matrix P, with a determinant of 1 that diagonalizes A__Make the change of coordinate system

**x=Px’.**This changes the equation to λ1x’^2 + λ2 y’^2 + λ3 z’^2 + jx’ + ky’ + lz” +m = 0λ1, λ2, and λ3 are the

__eigenvalues__of A

Complete the square to rewrite the equation of E into

**(x’’, y’’, z’’)**as the equation of a__quadric__in standard formUse the equation x’ = P^Tx to determine the centre and axes of E in the terms of the

__original__coordinate system.

**Ruled Surface in R^3**- a surface that can be made up from a family of __straight lines__

**x^2 + y^2 - z^2 = 1**The surface meets each horizontal plane in a circle whose

__centre lies on the z-axis__The surface meets each plane containing the z-axis in a

__rectangular hyperbola__**Family of Generators**- (generating lines) straight lines that__construct__the hyperboloid**Ruled surface**-__surface__created by the generating linesThere is a

__second family__of lines that are also generators of said surface, these lines are__rotated__around the z-axisThe lines that share a family never meet, but the lines of opposite families will always meet

**Theorem 2:** A hyperboloid of one sheet contains __two families__ of generating lines, **The members of each family are disjoint, and each member of either family intersects each member of the other,** with exactly __one exception__

Theorem 2 is also valid for this figure

**Conic section**- Shapes that are obtained by taking different plane slices through a __double cone__

__Menaechmus__ (Greek mathematician) discovered conic sections around __350 BC__

Slice 7 is considered an ellipse

**Ellipse-**An area where the sum of the distances from 2 fixed points on the plane remain constant

The

__ellipse__and the__hyperbola__both have a**centre****Non-degenerate conics**- conics that are__parabolas, ellipses, or hyperbolas__**Degenerate**- A single point, line, and pair of lines

**Circle**- a__type of conic,__a set of points that lie at a fixed distance (radius), from a fixed point (centre)Use the techniques of coordinate geometry to find the

__equation of a circle__with given centre and radius.**Theorem 1**: The equation of a circle with centre (a,b) and radius r is (x-a)^2 + (y-b)^2 = r^2Why?

This is a variation of the distance formula (d= (x1-x2)^2 + (y1-y2)^2)

Write the equation for the given centre and radius

(a) centre the origin, radius 1

(1)^2 - (x-0)^2 + (y-0)^2

1= x^2 + y^2

**x^2 + y^2 -1 = 0**

(b) centre (3, 4), radius 3

(3)^2 = (x-3)^2 + (y-4)^2

9= (x-3)(x-3)+ (y-4)(y-4)

9 = x^2 + y^2 -6x -8y + 25

**x^2 + y^2 -6x -8y +16=0**

Determine the condition on the numbers f, g, and h in the equation x^2 + y^2 + fx + gy + h = 0 for the equation to pass through the __origin__

This is the

__factored/ distributed form__of our original equationWe have to find the state of the equation when

**a and b equal zero**…. (x-0)^2 and (y-0)^2We can factor our this form to see the conditions

This would mean that

**f,g, and h all must equal 0**

(c) Consider the set of points (x,y) that satisfy the equation: x^2 + y^2 -4x +6y +9 = 0

Our goal is to complete the square (note that the coefficients on x^2 and y^2 are both 1)

Re-write equations containing just the x-terms, and just the y-terms

x^2 − 4x = (x − 2)^2− 4

y^2 + 6y = (y + 3)^2 − 9

These terms can be substituted into our original equation

(x-2)^2 -4 + (y+3)^2 -9 +9 = 0

(x-2)^2 + (y+3)^2 = 4

This means that the circle has a radius of 2 and a centre of (2, -3)

**Theorem 2**: You can __complete the square__ in the equation x^2 + y^2 + fx + gy + h = 0 to put it in the form (x+1/2f)^2 + (y+1/2g)^2 = 1/4 f^2 +1/4g^2 -h

This equation represents a circle with the centre (-1/2f, -1/2g) and a radius sqrt(1/4f^2 + 1/4g^2 -h)

This only occurs when the 1/4f^2 + 1/4g^2 -h is greater than zero, if this is less than zero, we can prove that the figure is __not a circle__

**Why?**

The equation is equal to r^2 (radius squared), when any number is squared, it has to equal a positive number. Also, when you go to take the square root of this side of the equation, it is impossible to get a rational answer when square-rooting a negative number.

(d) Determine the centre and radius of the equation

3x^2 + 3y^2 − 12x − 48y = 0.

(e) Determine the set of points that satisfies the following equation

x^2 + y^2 − 2x + 4y + 5 = 0;

Two intersecting circles that meet at __right angles__

**Theorem 3**: __Orthogonality__ test

Two intersecting circles with equations x^2 + y^2 + f1x + g1 y + h1 = 0 and x^2 + y^2 + f2x + g2 y + h2 = 0, are only orthogonal if f1 f2 + g1g2 = 2(h1 + h2).

Why?

This can be tested with the __pythagorem theorem__, where P is a point of intersection, and A and B are points inside of each of the circles. This means that

AP^2 + BP^2 = AB^2. Using our previous formulas, we know that

This can be solved to result in the original equation

In this image, the circle intersects at distinct points P and Q

This equation represents a circle when k __does not__ equal -1, and this means that P and Q will satisfy the equation

These circles have distinct points of intersection, and the line containing these points can be expressed using the second equation, only when __k=-1.__ When k is any other number, this is the equation of __one of the circles__

The three

__non-degenerate conics__are the parabola, ellipse, and hyperbolaThe distance of a set of points from a fixed point (

**focus**) follow a constant multiple (**eccentricity)**This distance is also followed from a line (

**directrix**)

**Eccentricity**- It is an ellipse if e (eccentricity) is between 0 and 1, a parabola of e equals 1, and a hyperbola is e is greater than 1

The set of points (P) whose distance from a fixed point (F) is equal to their distance from a fixed line (D)

Standard form:

The focus must lie on the x-axis, and have coordinates

**(a,0)**and a must be greater than 0The directrix must be

**x=-a**

y^2=4ax

This equation is used to show that each

__real number__corresponds with__one number__on the parabola

The focus and directrix can be described by

x = at^2, y = 2at (t ∈ R).

The x-axis is the axis of the parabola in standard form because the parabola is

__symmetric__in respect to this line.The

__origin__is the__vertex__of the parabola, and it has no centre

e is greater than or equal to zero, but less than one

A set of points whose distance from a certain point is e times their distance from a fixed line

Standard form:

The focus (point) must lie on the x-axis, and have coordinates

**(ae, 0)**, with a being greater than 0The directrix must follow the equation

**x=a/e**

This equation continues to grow, however, as the figure is

__symmetrical.__There is a second focus

**(-ae, 0)**and a second directrix**(-a/e)**The ellipse intersects the axes at the points

**(±a, 0) and (0, ±b).****Major axis**- the segment joining the points (±a, 0)**Minor axis-**the segment joining the points (0, ±b)The minor axis will always be shorter than the major axis because a is greater than b

Parametric equations:

x = a cost, y = b sin t (t ∈ (−π, π]).

e is greater than 1

Standard form:

The focus (point) must lie on the x-axis, and have coordinates

**(ae, 0),**with a being greater than 0The directrix must follow the equation

**x=a/e**

When: b^2 = a^2 (e^2 − 1), a > 0, e > 1.

Foci

**(±ae, 0)**and Directrices**x = ±a/e**Parametric equations: x = a sec t, y = b tan t (t ∈ (−π/2, π/2) ∪ (π/2, 3π/2)).

**Asymptotes-**the lines of the parabola that the branches grow closer to. They are notated as y = ±(b/a)x

e= sqrt (2)

This means that e^2=2 and a=b

The

__asymptotes__have the equation**y = ±x,**so they are at__right angles__A hyperbola whose asymptotes are at right angles is called a

__rectangular hyperbola__

You can use the asymptotes as new

__x and y axes,__and the equation can then be written as xy = c^2The origin will always be at the

__center__in this form

Parametric equations: x = ct, y = c/t where t = 0.

You can describe the equation of a non-degenerate conic in terms of

**polar coordinates**Polar coordinates: r and θ

This formula comes from the equation **OP = e · PM** where the distance from the origin to a point is equal to the eccentricity multiplied by the distance between that same point and the directrix.

**Theorem 5:** The sum of the two focal distances from a point always equal double the major axis.

Why?

The distance from a point to __either foci__ is equal to the __eccentricity times the distance from said point__ __to the directrix.__ This distance is equal to the distance from the second foci to to second directrix **(d’ and f’).** When adding these two distances, they will always equal **2a** (double the major axis)

**Theorem 6:** Differences of __Focal Distances__ of Hyperbola

If a point (P) is closer to

**F**, then**PF’ - PF = 2a**If P is closer to

**F’**then**PF’ - PF = -2a**|PF − PF| is

__constant__for every point on the parabola

A slant plane (pi) that cuts one portion of a right, circular cone is an __ellipse__

Goals: Find the slope of the line that is __tangent__ to the curve at the __parameter__, so that you can determine the __equation of the tangent__

**Theorem 1:** The slope of a tangent to a curve is **y’(t)/ x’(t)**

**Theorem 2:** Equations of tangents in __standard form__

**Polar** of (a,b) with respect to the unit circle

If (a,b) is some point on a tangent, then it can be determined that

**ax1 + by1 = 1**because of the equation**xx1 + yy1 = 1**. This equation can be re-written with points x2 and y2, which deduces the fact that points( x1, y1) and (x2, y2) are on the

__same line,__and this line is called the__polar of (a,b) with respect to the unit circle__

**Theorem 3:** When a point (a,b) lies __outside__ of the circle, and the tangents from point (a,b) touch the circle at P1 and P2, then the equation of the line is **ax + by = 1**

**Normal of a curve**- the line through a point that is__perpindicular to the tangent__of a curve at said pointTo find the equation of a normal of the curve, you find its

__slope__and the__coordinates__of the point on the parabola where they intersect

**Reflection law**- The angle that __incoming light__ makes with the __tangent__ to a surface is the same as the angle that the __reflected light__ makes with the tangent.

Applies to all mirrors, whether

__plane__or__curved__Mirrors are often designed to have a

__conic curve__

**Reflection property of the Ellipse** - light which comes from one focus of an __elliptical mirror__ is reflected at the ellipse to pass through the __second focus.__

**Reflection property of the hyperbola-** light coming from one focus of a hyperbolic mirror is reflected at the __hyperbola__ and makes the light appear to have come from the other focus (**Internal Reflection property**). Light going towards one focus is reflected towards the __other focus__ **(External Reflection Property**)

**Reflection Property of the Parabola**- incoming light __parallel__ to the axis is reflected at the __parabola__ to pass through the __focus__. Light coming from the focus of a parabola is reflected to give a beam of light parallel to the axis of the parabola

__Non-degenerate conics__are the__envelopes__of a family of lines that are tangent to the conicsThe conic being constructed is the

__curve in the plane__that has each of the lines in the family as a tangent

**Auxiliary circle**- A circle whose diameter is its major axisThe tangent to a parabola at its vertex is the auxiliary circle of the parabola

**Theorem 4:**A perpindicular from a focus of a non-degenerate conic to a tangent meets the tangent on the auxiliary circle of the conicWhen given a parabola and its axis, this theorem is used to identify the focus

**Parabola:**

**Ellipse:**

**Hyperbola:**

**Standard form of conics**- the centre is at the __origin__, and the axes are parallel to the __x and y axis__

Equations of all

__non-degenerate conics__follow this equation in__standard form:__Ax^2 + Bxy + Cy^2 + Fx + Gy + H = 0,

Any non-degenerate conic can be determined from a conic in standard form using a

__rotation__**(x, y) → (x cos θ − y sin θ, x sin θ + y cos θ)**followed by a__translation__**(x,y) → (x-a, y-b)**Both tranformations are

__linear__, so the equation of the conic at each stage is just a__second degree__of the original equation

Matrix form of the equation: **x^T Ax + J^T x + H = 0.**

Look back at the old equation: Ax^2 + Bxy + Cy^2 + Fx + Gy + H = 0

The values from this form are used to plug into the

__matrix form__

**Theorem 2:** A 2x2 matrix P represents a __rotation__ of R^2 when P is __orthogonal__ and the det of P = 1

Classify a conic (E) with the equation Ax^2 + Bxy + Cy^2 + Fx + Gy + H = 0

Write the equation in

__matrix__form (**x^T Ax + J^T x + H = 0**)Determine the

__orthogonal matrix__P, with a determinant of**1**that__diagonalizes__AMake the change of coordinate system

**x=Px’**. This changes the equation to**λ1x’^2 + λ2 y’^2 + f x’ + gy’ + h = 0**λ1 and λ2 are the

__eigenvalues__of A

Complete the square to rewrite the equation of E into

**(x’’, y’’)**as the equation of a conic in standard formUse the equation x’ = P^Tx to determine the

__centre__and__axes__of E in the terms of the__original__coordinate system.

**Quadric surfaces**- (quadrics) surfaces in R^3 that are the__natural analogues__of the conics.Ax^2 + By^2 + Cz^2 + Fxy + Gyz + Hxz + Jx + Ky + Lz + M = 0

**Degenerate quadrics**- Empty set, single point, single line, single plane, pair of planes, and a cylinder**Cylinder**- any surface that consists of an ellipse, parabola, or hyperbola in some plane

Use the

__orthogonal diagonalization__of 3x3 matrices to classify non-degenerate quadrics, similarly to how 2x2 matrices were used to__classify conics__

**Idetifying matrices:** In the equation Ax^2 + By^2 + Cz^2 + Fxy + Gyz + Hxz + Jx + Ky + Lz + M = 0,

If a 3x3 matrix (P) represents a __rotation__ of R^3 about the origin when __P is orthogonal__ and **det P = 1**

Classify quadric E with the equation Ax^2 + By^2 + Cz^2 + Fxy + Gyz + Hxz + Jx + Ky + Lz + M = 0,

Write the equation in matrix form (

**x^T Ax + J^T x + M = 0**)Determine the

__orthogonal matrix P, with a determinant of 1 that diagonalizes A__Make the change of coordinate system

**x=Px’.**This changes the equation to λ1x’^2 + λ2 y’^2 + λ3 z’^2 + jx’ + ky’ + lz” +m = 0λ1, λ2, and λ3 are the

__eigenvalues__of A

Complete the square to rewrite the equation of E into

**(x’’, y’’, z’’)**as the equation of a__quadric__in standard form__original__coordinate system.

**Ruled Surface in R^3**- a surface that can be made up from a family of __straight lines__

**x^2 + y^2 - z^2 = 1**The surface meets each horizontal plane in a circle whose

__centre lies on the z-axis__The surface meets each plane containing the z-axis in a

__rectangular hyperbola__**Family of Generators**- (generating lines) straight lines that__construct__the hyperboloid**Ruled surface**-__surface__created by the generating linesThere is a

__second family__of lines that are also generators of said surface, these lines are__rotated__around the z-axisThe lines that share a family never meet, but the lines of opposite families will always meet

**Theorem 2:** A hyperboloid of one sheet contains __two families__ of generating lines, **The members of each family are disjoint, and each member of either family intersects each member of the other,** with exactly __one exception__

Theorem 2 is also valid for this figure