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Homework Help: An Unknown Conic and Equation of Tangent Line HELP

  1. Jan 20, 2009 #1
    An Unknown Conic and Equation of Tangent Line **HELP**

    1. The problem statement, all variables and given/known data
    (a) Find the positive y-coordinate of the point on the conic whose x-coordinate is 27cos[tex]\theta[/tex], where 0<[tex]\theta[/tex]<[tex]\pi[/tex]/2
    (b) The slope of the tangent line to the conic through the point (27cos[tex]\theta[/tex], y1) is -1/27cot[tex]\theta[/tex], where y1 is your solution in part (a). Find the equation of the tangent line to the conic through the point (27cos[tex]\theta[/tex], y1)
    (c) Write an expression for S where S is the sum of the x- and y-intercepts of the tangent line that you computed in part (b)
    (d) If [tex]\theta[/tex] = tan-1(1/3), show that the value of S is 103/2

    2. Relevant equations

    x2 / 272 + y2 = 1


    3. The attempt at a solution
    okay sooo...
    because it has the + sign I think it is an Ellipse with the general equation of:
    (x-h)2/a2 + (y-k)2/b2 = 1

    BUT I graphed it using my calculator and it gave me an almost straight line, to do this I rearranged the equation to give:
    y = 1 - [tex]\sqrt{x^2/27^2}[/tex]

    for part (d)[tex]\theta[/tex] = tan-1(1/3) then tan[tex]\theta[/tex] = 1/3
    YA...I'm stuck:cry:
    PLEASE HELP Guys:blushing:
     
  2. jcsd
  3. Jan 20, 2009 #2
    Re: An Unknown Conic and Equation of Tangent Line **HELP**

    lol does no one want to help with this one or what?!:smile:
     
  4. Jan 20, 2009 #3

    Mark44

    Staff: Mentor

    Re: An Unknown Conic and Equation of Tangent Line **HELP**

    Is x^2/27^2 + y^2 = 1 the equation of the conic?

    I don't think you're giving us all the information.
     
  5. Jan 20, 2009 #4
    Re: An Unknown Conic and Equation of Tangent Line **HELP**

    Yes, Mark that is the equation of the conic given. The only part of the question I did not type out is the first part:
    You are living in a time just before Newton ad Leibniz. One night as you sleep, you have dream involving the conic:
    x2/272 + y2 = 1

    I didn't think it mattered lol
     
  6. Jan 20, 2009 #5

    Mark44

    Staff: Mentor

    Re: An Unknown Conic and Equation of Tangent Line **HELP**

    Yeah, it matters a lot!
    Since you're living before Newton and Leibniz, there is no calculus. The equation is that of an ellipse whose center is at (0, 0) and whose vertices are at (+/-27, 0) and (0, +/-1).

    If you solve x2/272 + y2 = 1 for y, you don't get what you did. Check your work and try again to find y_1.
    You are given the slope of the tangent line at (27 cos(theta), y_1). Hopefully you can find the equation of this line once you know a point on the line and its slope.
     
  7. Jan 22, 2009 #6
    Re: An Unknown Conic and Equation of Tangent Line **HELP**

    hi mark, I just cannot see how my y is wrong?! Here's what I did:
    (x^2/27^2) + y^2 = 1
    y^2 = 1 - (x^2/27^2)
    y =sqrt{ 1 - (x^2/27^2)}

    Would I now plug in 27cos(theta) for my x? But what is theta? All I know about the tangent line is y = (-1/27cot(theta))x + b --> to find b I habe to plug in the x = 27cos(theta) and the y that I haven't found right?
    thanks already
     
  8. Jan 22, 2009 #7
    Re: An Unknown Conic and Equation of Tangent Line **HELP**

    Well, pal.. i dont find a reason to make this thing so complicated, when it isnt! :).

    Go a bit systematically, you have been given the equation of the ellipse which is,
    [tex]\frac{x^2}{27^2} +\frac{y^2}{1^2}=~1[/tex]

    Now substitue, the value of x in the above given equation,
    thus, you have,
    [tex]\frac{(27cos\theta)^2}{27^2} +\frac{y^2}{1^2}=~1[/tex]
    now, cancelling [tex]27^2[/tex] from the numerator and the denomintor, you get,
    [tex]y^2=~1-cos^2\theta[/tex]
    Therefore, [tex]y=sin\theta[/tex]

    Now try solving the problem further, buddy!
     
  9. Jan 22, 2009 #8
    Re: An Unknown Conic and Equation of Tangent Line **HELP**

    hey you, thanks alot! So when I plug it into y=mx+b I get:
    sin(theta)=(-1/cot(theta))cos(theta) + b is this the answer to part (b)??
    what is (c) asking for here?!
    calculus is another language! :(
     
  10. Jan 22, 2009 #9
    Re: An Unknown Conic and Equation of Tangent Line **HELP**

    Rather, the solution to (b) would be,
    [tex]sin\theta=~\frac{-1}{27cot\theta}.27cos\theta +~b[/tex]
    Therefore,
    [tex]sin\theta=~\frac{-sin\theta}{27cos\theta}.27cos\theta +~b[/tex]

    Thus,
    27cos\theta gets cancelled and the value of b turns out to be [tex]2sin\theta[/tex]
    Therefore, the equation would be,
    [tex]27ycos\theta +~xsin\theta=~2sin\theta [/tex]
    (Do try it once again!)
     
  11. Jan 22, 2009 #10
    Re: An Unknown Conic and Equation of Tangent Line **HELP**

    Oh I see what you did there, you converted cot to sin/cos...genius :D
    so now this is the equation for the tangent line, awesome...next problem, what am I supposed to do for part (c), what is the question asking for? the Sum of the x and y intercepts???
     
  12. Jan 22, 2009 #11
    Re: An Unknown Conic and Equation of Tangent Line **HELP**

    Pal, I've helped you a lot with this, you need to have your own input over it, give it a try!
     
  13. Jan 22, 2009 #12
    Re: An Unknown Conic and Equation of Tangent Line **HELP**

    :cry:could you maybe just reword it or something? sorry I know you've helped me loads but it seems to be hard for me understand what the question is asking for:frown: I'm german :redface:
     
  14. Jan 22, 2009 #13
    Re: An Unknown Conic and Equation of Tangent Line **HELP**

    Well, please check the question once! I'm facing a petty problem..
     
  15. Jan 22, 2009 #14
    Re: An Unknown Conic and Equation of Tangent Line **HELP**

    Uhm, okay so I just wrote the entire question out again and tried to find out what 'the sum of the x and y intercepts' means but I cannot get any further with my problem :(
     
  16. Jan 22, 2009 #15

    Mark44

    Staff: Mentor

    Re: An Unknown Conic and Equation of Tangent Line **HELP**

    You have the equation of the tangent line.
    1. Find the x-intercept of this line.
    2. Find the y-intercept of this line.
    3. Add the two intercepts together.
     
  17. Jan 22, 2009 #16
    Re: An Unknown Conic and Equation of Tangent Line **HELP**


    OKAY so:
    27ycos[tex]\theta[/tex] = -(sin[tex]\theta[/tex])x + 2sin[tex]\theta[/tex]

    x-intercept:
    let y=0
    0 = -sin[tex]\theta[/tex]x + 2sin[tex]\theta[/tex]
    (sin[tex]\theta[/tex])x + 2sin[tex]\theta[/tex]
    x = (2sin[tex]\theta[/tex])/sin[tex]\theta[/tex]


    y-intercept:
    let x=0
    27ycos[tex]\theta[/tex] = 0 + 2sin[tex]\theta[/tex]
    27ycos[tex]\theta[/tex] = 2sin[tex]\theta[/tex]
    y = 2sin[tex]\theta[/tex]/27cos[tex]\theta[/tex]


    Sum of x- & y-intercepts:
    S = 2sin[tex]\theta[/tex]/sin[tex]\theta[/tex] + 2sin[tex]\theta[/tex]/27cos[tex]\theta[/tex]
    S = 2 + (2/27)tan[tex]\theta[/tex]


    for part (d):
    [tex]\theta[/tex] = tan-11/3
    I will now plug this in to S from part (c):
    S = 2 + (2/27)tan(tan-11/3
    S = 2.025
    WRONG --> the answer is supposed to be 102/3 which is 31.62 ????
    What is wrong here? I don't understand??:confused:
     
  18. Jan 22, 2009 #17
    Re: An Unknown Conic and Equation of Tangent Line **HELP**

    You are so wrong for part c and d, wait for Prof Sandeep/Sarkar to post up the solutions in clew.
     
  19. Jan 22, 2009 #18
    Re: An Unknown Conic and Equation of Tangent Line **HELP**

    lmao Okay, MrUnknown who is either Justin or someone who knows Justin...that would be alittle late dont you think?...why dont you tell me the right answer then?
     
    Last edited: Jan 22, 2009
  20. Jan 22, 2009 #19
    Re: An Unknown Conic and Equation of Tangent Line **HELP**

    Yes, thats what Anna, I could not get to [tex]10^{3/2}[/tex], which is why i asked you check the question! As of finding the intercepts, all looks good to me!
     
  21. Jan 22, 2009 #20
    Re: An Unknown Conic and Equation of Tangent Line **HELP**

    o wow that sucks...a lot i just checked, i typed everything right...no mistakes :(
     
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