parametric equations: showing tangent to curve is.....

hi, doing a questiong on parametric equations. i'll tell you what i have done so far.

x=3cos(theta) y=2sin(theta) for 0</= (theta) <2pi

i) i had to eliminate theta to form cartesian equations. i got x^2/9 + y^2/4 = 1

ii) had to draw the curve. it is an elipse which cuts through y= 2 and -2 and x= 3 and -3. i had to drew tangents through point (6,2).

iii) i had to use parametric equations to calculate dy/dx in terms of theta.

- 2cos (theta) / 3sin (theta)

this is where im stuck, although i said that yesterday and seemed to know where i went wrong afterwards.

so now i'm given that the equation of the tangent to the curve (3cos (theta), 2sin (theta)) is 2x cos (theta) + 3y sin (theta) = 6

i have to show that for the tangents to the curve which passes through (6,2),2cos (theta) + sin (theta) =1

i have another part to do afterwards but i'll have a go at it before i ask again.

Reply 1

*bobo*

8

apply x=6 and y=2 to 2xcos theta + 3ysintheta=6
which gives 12costheta + 6sintheta=6
=> 2costheta + sintheta=1

Reply 2

jones001

OP

oh thats what i did....but i didnt cancel it down =[ thanx :smile:

Reply 3

jones001

OP

kk iv done one part to this but not getting the right answer for the other part., it says, solve equation 2cos (theta) + sin (theta)=1 to find two values of theta in radians to 2 dp, corresponding to the two tangents.

2cos(theta) + sin (theta) =1

2cos (theta) =0

sin (theta)=1
theta=pi (1.57)

this is right but the next one im doing is wrong, by 0.4

sin (theta)=0

2cos(theta)=1
cos (theta)=1/2
(theta)=1.047197551

using "all-silly-tom-cats"

as it was cos, i did 2pi-1.04719551 and got 5.243185307, this case 5.24, but the answer is meant to be 5.64, am i wrong or is it the book?

Reply 4

Trangulor

6

2cos(theta) + sin (theta) =1

2cos (theta) =0

sin (theta)=1
theta=pi (1.57)

That was fortunate. You could do that by sight straight away (and by looking at your sketch), but your method here is wrong.

sin (theta)=0

2cos(theta)=1
cos (theta)=1/2
(theta)=1.047197551

The two bold lines imply different values of theta. Try putting 1.047197551 into the equation to see if you get 1.

Do you know how to convert something like '2cos(theta) + sin (theta)' into Rcos(theta+alpha), where R and alpha are constants?