## 1ace 2 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. 1ace

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

4. 1ace

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

5. 1ace

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

6. 1ace

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. 1ace

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. 1ace

OOOHH omg hahaahah thats where i went wrong?!

15. 1ace

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. 1ace

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. 1ace

okay!

22. freckles

Right!: )

23. freckles

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

24. freckles

$2x-2-200x=0$

25. 1ace

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

26. freckles

No no.

27. 1ace

why not ?

28. freckles

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

29. 1ace

ok, so i get x=1/98 ?

30. 1ace

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

31. 1ace

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. 1ace

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. 1ace

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. 1ace

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

42. freckles

|dw:1353821881622:dw|

43. 1ace

thank you !! i get it now !!

44. freckles

Np. :)