anonymous one year ago The slope of the tangent to a curve at any point (x, y) on the curve is -x/y. Find the equation of the curve if the point (2, −2) is on the curve.

1. anonymous

not sure how to do this one

2. ganeshie8

what do you know about the relation between slope of tangent line and first derivative ?

3. anonymous

well the first derivative represents the slope of the tangent line on any point of the function you derive

4. ganeshie8

You're given that the slope of tangent line equals -x/y, so setup an equation using that info

5. anonymous

i thought that -x/y was the derivative

6. ganeshie8

Yes $\dfrac{dy}{dx}=\dfrac{-x}{y}$

7. anonymous

what do i solve for though ?

8. ganeshie8

thats the equation ^

9. ganeshie8

you need to solve the curve $$y$$

10. anonymous

so implicit differentiation ?

11. ganeshie8

An usual algebraic equation involves isolating a "variable", but a differential equation involves solving for a "curve"

12. ganeshie8

familiar with variable separation ?

13. anonymous

yep, give me a sec

14. ganeshie8
15. anonymous

okay im getting $\sqrt{C - x^2}$

16. anonymous

is that right?

17. ganeshie8

looks partially correct, show me ur work

18. anonymous

|dw:1439277292623:dw|

19. anonymous

this drawing tool is dreadful

20. anonymous

then i isolated y

21. ganeshie8

leave it as $y^2=-x^2+C$

22. anonymous

|dw:1439277527370:dw|

23. ganeshie8

Now plugin the initial condition, $$(2, -2)$$, and solve $$C$$

24. ganeshie8

|dw:1439277677058:dw|

25. anonymous

|dw:1439277671100:dw|

26. ganeshie8

|dw:1439277771874:dw|

27. anonymous

oh woops , my mind was slipping

28. anonymous

okay so know we just plug that into |dw:1439277880677:dw| right?

29. anonymous

these area my answer choices by the way x + y = 0 x2 − y2 = −2 x2 + y2 = 16 x2 + y2 = 8 i assume its D

30. ganeshie8

Yes

31. anonymous

thanks for the help

32. ganeshie8

$y^2=-x^2+C$ plugin $$(2, -2)$$ $(-2)^2=-2^2+C\implies C = 8$ so the solution is $y^2=-x^2+8$ which is same as $x^2+y^2=8$