## anonymous 3 years ago Find the points on the ellipse 100x^2+y^2=100 that are farthest away from the point (1,0)

1. freckles

You know the distance formula?

2. freckles

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

3. anonymous

ok i know y^2=(100-100x^2)

4. anonymous

so d^2=x^2 + (100-100x^2)^2

5. anonymous

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

6. anonymous

or am I just doing this all wrong? :\

7. freckles

$d=\sqrt{(\Delta x)^2+(\Delta y)^2}=\sqrt{((x-1)^2+(y-0)^2}=\sqrt{(x-1)^2+y^2}$ I used the points given (1,0) and (x,y)

8. freckles

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

9. freckles

$d^2=(x-1)^2+y^2$ Replace y^2 with what you found :)

10. anonymous

yep i did that ! :o

11. freckles

$100x^2+y^2=100 => y^2=100-100x^2$

12. freckles

Why did you square y^2?

13. freckles

$d^2=(x-1)^2+100-100x^2$

14. anonymous

OOOHH omg hahaahah thats where i went wrong?!

15. anonymous

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

16. freckles

Nope nope.... So when you found ((x-1)^2)' You used chain rule right?

17. freckles

We know that (100)'=0

18. freckles

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

19. anonymous

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

20. freckles

You need to distribute that 2. 2(x-1)=2x-1(2)=2x-2

21. anonymous

okay!

22. freckles

Right!: )

23. freckles

$\text{ Set } (d^2)'=0$

24. freckles

$2x-2-200x=0$

25. anonymous

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

26. freckles

No no.

27. anonymous

why not ?

28. freckles

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

29. anonymous

ok, so i get x=1/98 ?

30. anonymous

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

31. anonymous

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

32. freckles

$y=\pm \sqrt{100-100x^2} \text{ correct?}$

33. freckles

Say yes. lol

34. anonymous

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

35. freckles

$(\frac{1}{99}, \sqrt{100-100(\frac{1}{99})^2}) \text{ and } (\frac{1}{99},-\sqrt{100-100(\frac{1}{99})^2})$

36. anonymous

okay but why 1/99? i thought we found that 2x-2-200x=0 then wouldnt x be 2/-198 ?

37. freckles

|dw:1353821740124:dw|

38. freckles

$198/2=99$

39. freckles

$2/198=1/99$

40. freckles

|dw:1353821859781:dw|

41. anonymous

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

42. freckles

|dw:1353821881622:dw|

43. anonymous

thank you !! i get it now !!

44. freckles

Np. :)