1ace
Find the points on the ellipse 100x^2+y^2=100 that are farthest away from the point (1,0)



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freckles
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You know the distance formula?

freckles
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You want to maximize that using the points (1,0) and (x,y)

1ace
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ok i know y^2=(100100x^2)

1ace
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so d^2=x^2 + (100100x^2)^2

1ace
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but im not sure about the x^2 part is it (1x)^2 because the point is (1,0)?

1ace
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or am I just doing this all wrong? :\

freckles
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\[d=\sqrt{(\Delta x)^2+(\Delta y)^2}=\sqrt{((x1)^2+(y0)^2}=\sqrt{(x1)^2+y^2}\]
I used the points given (1,0) and (x,y)

freckles
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Okay and you squared the distance!
Good job! :)

freckles
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\[d^2=(x1)^2+y^2\]
Replace y^2 with what you found :)

1ace
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yep i did that ! :o

freckles
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\[100x^2+y^2=100 => y^2=100100x^2\]

freckles
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Why did you square y^2?

freckles
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\[d^2=(x1)^2+100100x^2\]

1ace
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OOOHH omg hahaahah thats where i went wrong?!

1ace
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so then after i take the derivative is do i find the roots ?
199x1=0? that doesnt seem right :s

freckles
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Nope nope....
So when you found ((x1)^2)'
You used chain rule right?

freckles
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We know that (100)'=0

freckles
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(100x^2)'=100(x^2)'
we know to use power rule here

1ace
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yes so i get 2(x1)*1 + 0 200x

freckles
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You need to distribute that 2.
2(x1)=2x1(2)=2x2

1ace
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okay!

freckles
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Right!: )

freckles
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\[\text{ Set } (d^2)'=0\]

freckles
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\[2x2200x=0\]

1ace
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ok but dont i have to divide by the 2d from when i differentiate (d^2)'

freckles
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No no.

1ace
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why not ?

freckles
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Minimizing d^2 will give us the same results as minimizing d

1ace
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ok, so i get x=1/98 ?

1ace
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then would that be my point? x= 1/98 and subbing that into the original equation y= 9.99948?

1ace
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also i only get 1 answer, shouldnt there be another point also?

freckles
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\[y=\pm \sqrt{100100x^2} \text{ correct?}\]

freckles
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Say yes. lol

1ace
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but my x is only 1 point? and my y's will be 2 points? sorry im just a little confused

freckles
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\[(\frac{1}{99}, \sqrt{100100(\frac{1}{99})^2}) \text{ and } (\frac{1}{99},\sqrt{100100(\frac{1}{99})^2})\]

1ace
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okay but why 1/99? i thought we found that 2x2200x=0
then wouldnt x be 2/198 ?

freckles
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dw:1353821740124:dw

freckles
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\[198/2=99\]

freckles
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\[2/198=1/99\]

freckles
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dw:1353821859781:dw

1ace
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omg ok i see !!! i think i got it now !!!

freckles
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dw:1353821881622:dw

1ace
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thank you !! i get it now !!

freckles
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Np. :)