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

1. Jan 20, 2009

### AnnaSuxCalc

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$$\theta$$, where 0<$$\theta$$<$$\pi$$/2
(b) The slope of the tangent line to the conic through the point (27cos$$\theta$$, y1) is -1/27cot$$\theta$$, where y1 is your solution in part (a). Find the equation of the tangent line to the conic through the point (27cos$$\theta$$, 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 $$\theta$$ = 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 - $$\sqrt{x^2/27^2}$$

for part (d)$$\theta$$ = tan-1(1/3) then tan$$\theta$$ = 1/3
YA...I'm stuck

2. Jan 20, 2009

### AnnaSuxCalc

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

lol does no one want to help with this one or what?!

3. Jan 20, 2009

### 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.

4. Jan 20, 2009

### AnnaSuxCalc

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

5. Jan 20, 2009

### 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.

6. Jan 22, 2009

### AnnaSuxCalc

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?

7. Jan 22, 2009

### psykatic

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,
$$\frac{x^2}{27^2} +\frac{y^2}{1^2}=~1$$

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

Now try solving the problem further, buddy!

8. Jan 22, 2009

### AnnaSuxCalc

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! :(

9. Jan 22, 2009

### psykatic

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

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

Thus,
27cos\theta gets cancelled and the value of b turns out to be $$2sin\theta$$
Therefore, the equation would be,
$$27ycos\theta +~xsin\theta=~2sin\theta$$
(Do try it once again!)

10. Jan 22, 2009

### AnnaSuxCalc

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???

11. Jan 22, 2009

### psykatic

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!

12. Jan 22, 2009

### AnnaSuxCalc

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

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 I'm german

13. Jan 22, 2009

### psykatic

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

Well, please check the question once! I'm facing a petty problem..

14. Jan 22, 2009

### AnnaSuxCalc

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 :(

15. Jan 22, 2009

### 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.

16. Jan 22, 2009

### AnnaSuxCalc

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

OKAY so:
27ycos$$\theta$$ = -(sin$$\theta$$)x + 2sin$$\theta$$

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

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

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

for part (d):
$$\theta$$ = 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??

17. Jan 22, 2009

### Mughni

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.

18. Jan 22, 2009

### AnnaSuxCalc

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
19. Jan 22, 2009

### psykatic

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

Yes, thats what Anna, I could not get to $$10^{3/2}$$, which is why i asked you check the question! As of finding the intercepts, all looks good to me!

20. Jan 22, 2009

### AnnaSuxCalc

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 :(