Analytic Geometry

Conics and Nonlinear Systems of Equations

What to do … what to learn …. Illustrate the different types of conicsections: parabola, ellipse, circle,hyperbola, and degenerate cases

Determine the standard form of equation ofeach conic

Recognize the equation and importantcharacteristics of the different types ofconic sections

What to do … what to learn …. Illustrate systems of nonlinear equations Determine the solutions of systems ofnonlinear equations using techniques such assubstitution, elimination, and graphing

Solve situational problems involving conicsections and systems of nonlinear equations

The CONICS

CONICS FROM CONES

Conic sections are four shapes; parabolas, circles, ellipses, and hyperbolas, created from theintersection of a plane with a cone or two cones.

DOUBLE-NAPPED CONE

We use this solid ingenerating what we callconic sections or simplycones.

CONICS FROM CONES

THE CIRCLE

Definition A circle is the set ofall points in a planewhose distances from afixed point is aconstant. The fixedpoint is called thecenter of the circle andthe constant is calledthe radius of thecircle.

STANDARD EQUATION OF A CIRCLE

EXAMPLE 1 Write the standard form of equation of acircle having a diameter whose endpoints are(1,-4) and (3,10). Graph the circle.

EXAMPLE 1

EXAMPLE 2 Find the center and radius of the circlehaving equation

x2 + y2 + 2x – 6y + 1 = 0.

DEGENERATE CASE

KNOWLEDGE CHECK

Do the following exercises.A. Graph each of the following circles.

1. x2 + y2 = 42. (x – 4)2 + (y + 3)2 = 163. x2 – 6 x + y2 – 2y = 434. x2 + y2 + 2x – 6y + 7 =05. x2 + 2 x + y2 + 6y + 10 = 0

Do the following exercises.B. Find an equation of the circle satisfyingthe given condition.

11. Center (4,-5) and radius 312. Center (-2,-2) and radius 713. Center is at the origin, passing through(3,-4) 14. Center (-1,5), passing through (-4,-6)15. Endpoints of a diameter (7,-5), (-1,3)

The parabola

Definition A parabola is the set of all points (x,y) ina plane that are equidistant from a fixedline, called the directrix, and a fixedpoint called the focus, not on the line. Thevertex is the midpoint between the focus andthe directrix. The axis of the parabola isthe line passing through the focus and thevertex.

Illustration

Illustration

Parabolas in real life

Parabolas in real life

Do you know that ….The parabola is usedin the design of carheadlights and inspotlights because itaids in concentratingthe light beam?

Do you know that ….A parabolic reflectorhas the property thatif a light source isplaced at the focus ofthe reflector, thelight rays willreflect from themirror as raysparallel to the axis.

Do you know that ….For this reason,this is used in autoheadlights to givean intenseconcentrated beam oflight.

DefinitionA parabola is the set ofall points (x,y) in aplane that areequidistant from a fixedline, called thedirectrix, and a fixedpoint called the focus,not on the line.

Definition The vertex is themidpoint between thefocus and thedirectrix. The axis of the parabola isthe line passingthrough the focus andthe vertex.

Standard Equations

Standard Equations When the equation of the parabola is

(x – h)2 = 4p (y – k)

its vertex is the point (h,k); its focus is(h,k+p).

The parabola opens upward if p>0 and opensdownward if p<0.

Standard Equations When the equation of the parabola is

(y – k)2 = 4p (x – h)

its vertex is the point (h,k); its focus is(h+p,k).

The parabola opens to the right if p>0 andopens to the left if p<0.

Standard Equation: (x – h)2 = 4p (y – k)

Standard Equation: (y – k)2 = 4p (x – h)

Standard Equation: (x – 0)2 = 1/3 (y – 0)

Standard Equation:

(x – 0)2 = 1/3 (y – 0)

Standard Equation:

(x – 0)2 = 1/3 (y – 0)

Standard Equation:

(x – 0)2 = 1/3 (y – 0)

Standard Equation:

(x – 0)2 = 1/3 (y – 0)

Standard Equation: (x – 2)2 = –1/3 (y – 4)

Standard Equation: (x – 2)2 = –1/3 (y – 4)

Standard Equation: (x – 2)2 = –1/3 (y – 4)

Standard Equation: (x – 2)2 = –1/3 (y – 4)

EXAMPLE 1

Find the equation of the parabola withvertex at (0,0) and focus at (0,-5).

EXAMPLE 1

EXAMPLE 1

Find the equation of the parabola withvertex at (0,0) and focus at (0,-5).

Vertex (h,k) = (0,0)Focus (0,-5) is located 5 units below thevertex → p = -5.

EXAMPLE 1

The equation should take the form

(x – h)2 = 4p (y – k).

(h,k) = (0,0), p = -5

EXAMPLE 1 h = 0, k = 0, p = − 5

(x – h)2 = 4p (y – k).

(x – 0)2 = 4(-5) (y – 0)

or x2 = −20y.

EXAMPLE 1 x2 = −20y

EXAMPLE 2Reduce the following equation of a parabolato standard form.

y2 −6y – 8x + 1 = 0

EXAMPLE 2 (y – 3)2 = 8(x + 1)

summary

summary

EXAMPLE 3

Find an equation of the parabola withvertex at (-2,3) and focus at (0,3). Graphthe parabola.

EXAMPLE 3Find an equation of the parabola withvertex at (-2,3) and focus at (0,3). Graphthe parabola.

What is the value of h? Of k? What is the distance (p) between thefocus and the vertex?

What standard form of equation should beapplied here?

EXAMPLE 3

EXAMPLE 4Find an equation of the parabola with focusat (0,4) and directrix y = -4. Graph theparabola.

EXAMPLE 4Find an equation of the parabola with focusat (0,4) and directrix y = -4. Graph theparabola.

How do we compute for the vertex of theparabola? What is the vertex?

Which standard equation should we usehere?

What is the value of p?

EXAMPLE 4

Focal chord (or latus rectum)A focal chord (or latus rectum) is a

segment drawn through the focus and

parallel to the directrix so that it will

intersect the parabola at two points that

lie 2 p distance units from the focus.

Focal chord (or latus rectum)The length of the focal chord is 4 p.

Focal chord (or latus rectum)

EXAMPLE 5Find an equation of the parabola withvertex (0,4) and focus (0,6). Draw itsgraph and locate the ends of its focalchord.

EXAMPLE 5

Degenerate Case of Parabolas

KNOWLEDGE CHECK

Do the following Exercises.Item A #1 - #3

Item B #4 - #7

TIME TO REFLECT

THE ELLIPSE

Definition An ellipse is theset of all pointsin a plane, the sumof whose distancesfrom two distinctfixed points,called foci, isconstant.

The line through thefoci intersects theellipse at twopoints calledvertices. The chordjoining the verticesis the major axis,

and its midpoint isthe center of theellipse. The chordperpendicular tothe major axis atthe center is theminor axis of theellipse.

THE HYPERBOLA

Definition A parabola is the set of all points (x,y) ina plane that are equidistant from a fixedline, called the directrix, and a fixedpoint called the focus, not on the line. Thevertex is the midpoint between the focus andthe directrix. The axis of the parabola isthe line passing through the focus and thevertex.

WE CAN ALSO SEE THSES CONICS IN DESIGNS.

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