## silverxx Group Title find the tangents of the curve x^2 + y^2 - 6x + 4y = 0 from (-17, 7) the answers are 2x + 3y + 13 and 34x + 129y = 325 11 months ago 11 months ago

1. myininaya Group Title

You can find y'? I think you mean find the tangent of the curve at (-17,7), right? You can verify that point is on the circle by pluggin it in.

2. naranjja Group Title

$x^2+y^2-6x+4y=0$Deriving implicitly,$2x+2y*y'-6+4*y'=0$$2y*y'+4*y'=-2x+6$$y'(2y+4)=-2x+6$$y'=\frac{ 2x+6 }{ 2y+4 }$Evaluating at the point (-17,7),$y'=\frac{ 2(-17)+6 }{2(7)+4 }=-\frac{ 28 }{ 18 }=\frac{ 14 }{ 9 }$This is the slope. Now, using point-slope form,$y-7=\frac{ 14 }{9 }(x+17)$$y=\frac{ 14 }{ 9 }x+\frac{ 301 }{ 9 }$And that is the tangent that goes through that point.

3. silverxx Group Title

yes dy/dx = -2x+6/2y + 4

4. myininaya Group Title

hmmm...I don't think that point in on the circle.

5. silverxx Group Title

@naranjja you missed -2x + 6.. i think. the sign

6. naranjja Group Title

Damn, you're right.

7. silverxx Group Title

@myininaya i don't know.. hmm

8. naranjja Group Title

$y'=\frac{ -2x+6 }{ 2y+4 }$$y'=\frac{ -2(-17)+6 }{ 2(7)+4 }=\frac{ 40 }{ 18 }=\frac{ 20 }{ 9 }$Therefore,$y-7=\frac{ 20 }{ 9 }(x+7)$$y=\frac{ 20 }{ 9 }x+\frac{ 203 }{ 9 }$

9. myininaya Group Title

You want to find the tangent to the curve at (-17,7) That point isn't on the circle. Or do you want to find a tangent line to a point on the circle that goes through that point?

10. silverxx Group Title

you think it'll satisfy the answers?? it was written in the book hehe

11. silverxx Group Title

@myininaya the second

12. naranjja Group Title

And I wrote -7 instead of -17, my bad:$y-7=\frac{ 20 }{ 9 }(x+17)$$y=\frac{ 20 }{ 9 }x+\frac{ 403 }{ 9 }$

13. silverxx Group Title

@naranjja there are 2 answers hehe

14. myininaya Group Title

Is there anymore info given in the problem? Like the y-intercepts of the lines?

15. myininaya Group Title

Oh! Do you mean find the tangent points? And those are the lines that are given in the problem?

16. silverxx Group Title

yes yes

17. myininaya Group Title

"find the tangents of the curve x^2 + y^2 - 6x + 4y = 0 from (-17, 7) the answers are 2x + 3y =13 and 34x + 129y = 325" So we have 2x+3y=13 34x+129y=325 Put both into y=mx+b form. :)

18. myininaya Group Title

You guys already found the general slope m=y'=(-x+3)/(y+2)

19. silverxx Group Title

i need to find the process in order to reach the correct answers

20. silverxx Group Title

find's not the term but to solve

21. myininaya Group Title

Right. We are trying to find the tangent points. I'm telling you how to get there.

22. myininaya Group Title

You need to put both of those lines that were given into y=mx+b form.

23. myininaya Group Title

And is that one line 2x+3y=-13?

24. silverxx Group Title

yes

25. myininaya Group Title

I think it is because (-17,7) has to be on it. Ok so you find the slopes of both of those lines. Set it equal to the general slope of the curve you guys found. And solve both equations.

26. silverxx Group Title

wait.. my solution's -2x+6/ 2y+4 = y-7/ x- 17

27. myininaya Group Title

So we have 2x+3y=-13 34x+129y=325 I'm asking you to put both of those into y=mx+b form so we can identify the slopes of the tangent lines.

28. silverxx Group Title

got 2y^2 + 2x^2 - 40x - 10y + 79

29. silverxx Group Title

i think i have my own way to solve this..

30. silverxx Group Title

but i was just too stupid in not deriving the answers

31. myininaya Group Title

look I will do the first one for you it isn't hard We have 2x+3y=-13 Solve for y 3y=-2x-13 y=-2x/3-13 Slope is -2/3 So this means -2/3=(-x+3)/(y+2) Remember (-x+3)/(x+2) is what you guys found above. I just reduced the fraction. Solve for x and y. -2=-x+3 3=y+2 You should get point on a circle. More importantly the tangent point that is on the line 2x+3y=-13 that goes through (-17,7).

32. silverxx Group Title

okay :)

33. myininaya Group Title

But actually I could be wrong...ERRR...That result isn't on the circle. Let me think a little more.

34. myininaya Group Title

oh it is i just don't know how to add :)

35. myininaya Group Title

So that is one point. You can find the other using that same process.

36. myininaya Group Title

With the other line that was given to you.

37. silverxx Group Title

okay..

38. myininaya Group Title

34x + 129y = 325 So first step is to write this into y=mx+b form Identify the slope. Put y'=m and solve for x and y to find the other tangent point.

39. silverxx Group Title

okay. i'll try it later.. i think i need to digest this..

40. myininaya Group Title

You should get a result that isn't even on the circle. However you could find another tangent point whose line that goes through (-17,7) by looking at my workings from above.

41. silverxx Group Title

thank thank thank you so much @myininaya :))

42. myininaya Group Title

You can get another tangent point by looking at this: -2/3=(-x+3)/(y+2) I did -2=-x+3 , 3=y+2 You have also looked at it as 2/-3=(-x+3)/(y+2) which means you could solve 2=-x+3 and -3=y+2 to find the other point.

43. myininaya Group Title

But okay. I understand. Have fun.