### Conics: The Parabola, Circle and Ellipse (Cheek's 3rd Scribe post)

Hello its cheeks again working on today's scribe post on Conics: The Parabola, Circle and Ellipse. We started off by talking about our website reaching its max on 300 mb due the cause of all the images we posted up. So for now on we will be using photobucket to upload our images into our blog. We then watched some videos that were quite hilarious based and mathematics in relation with the real world. Our final video was about a tire then went loose as the driver was driving and how it rolled, flyed, bumped into other cars and reaching its final destination back into the drivers car again and the probability of that was 0.0000001%. Think its possible? Comment it. Our teacher then wanted all of us to comment about how mathematics is used in our other subjects. Think it is? Comment that too.

We continued our lesson where we left off from the anatomy of the parabola...

Our lesson began and ended with our dictionaries:__Deriving the standard form for the equation of a parabola__

__ __

PF=PD (by def'n)

√[(x-h)²+[y-(k+p)]]² = √[(x-x)²+[y-(k-p)]]² (Distance formula)

(x-h)²+[y-k-p][y-k-p] = [y-k+p][y-k+p] (Square both and simplify)

(x-h)²+y²-ky-py-ky+k²+kp-py+kp+p² = y²-ky+py-ky+k²-kp+py-kp+p² (expan)

if we were to bring the terms that cancel from the left to the right side we simplify to:

(x-h)² = 4py - 4pk (balance)

(x-h)² = 4p(y-k) (factored)

***Standard form for equation of a vertical formula

__ __

PF=PD (by def'n)

√[[x-(h+p)]²+(y-k)²] = √[[x-(h-p)]²+(y-k)²] (Distance formula)

[x-h-p][x-h-p] +(y-k)² = [x-h+p][x-h+p] (Square both and simplify)

(y-k)²+x²-hx-px-hx+h²+hp-px+hp+p² = x²-hx+xp-xh+h²-hp+px-ph+p² (expan)

if we were to bring the terms that cancel from the left to the right side we simplify to:

(y-k)² = 4px - 4ph (balance)

(y-k)² = 4p(x-h) (factored)

***Standard form for equation of a horizontal formula

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Circle: A locus of points that move such that they are a fixed distance (radius) from a fixed point (centre).__Deriving the standard form for the equation of a circle__

__

OP = r (by def'n)

√[(x-h)²+(y-k)²] = r (Distance formula)

(x-h)²+(y-k)² = r² (standard form for a circle)

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Ellipse: A locus of points that move such that the sum of the distances (focal radii) from fixed points (foci) is constant.

The anatomy of the ellipse

A_{1 } and A_{2} are the vertices of the ellipse

A_{1}A_{2} is the major axis of the ellipse. It is a line of symmetry and its length is 2a

B_{1}B_{2 are the endpoints of the minor axis.}B_{1}B_{2 is the minor axis of ellipse. It is a line of symmetry and its length is 2b.}_{F1 and F2 are teh foci of the ellipse each focus is C units from the centre.PF1 and PF2 are ca}_{lled the focus radii. Their sum is always 2a, the length of the major axis.}__The standard form for the equation of an ____ellipse__

Standard form for a horizontal equation

Standard form for a vertical equation

The focal radii property

Suppose P is at A

PF_{1} + PF_{2} = A_{1}F_{1} + A_{1}F_{2 } (by def'n)

PF_{1} + PF_{2} = A_{2}F_{2} + A_{1}F_{2 } (by symmetry

PF_{1} + PF_{2} = A_{1} A_{2}

PF_{1} + PF_{2} = 2a

The Pythagorean property

Suppose P is at B

PF_{1} + PF_{2} = B_{1}F_{1} + B_{1}F_{2 }

By symmetry we know that

B_{1}F_{2} = B_{1}F_{1}

there fore...

PF_{1} + PF_{2} = B_{1}F_{1} + B_{1}F_{1 }

PF_{1} + PF_{2} = 2B_{1}F_{1}

By the focal radii property

PF_{1} + PF_{2} = 2a

therefore..

2a = 2B_{1}F_{1}

a = B_{1}F_{1}

and by Pythagorean theorm

b² + c² = a² or c² = a² - b²

Thats all for tonight the next scribe is Jefferson

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