Trouble with Locus questions in Conics (1 Viewer)

[karnbmx]

Hey guys!

I have been having serious trouble with certain types of Conics questions, especially those that involve finding the locus of a point (especially those involving sine and cosine functions). Is there a good tip/trick to solving those types of problems?

 

[nightweaver066]

Use , and extensively.

 

[barbernator]

my tip would be go back to basics, and write down step by step what you know you have to do.
Also, in the HSC exam, they can only ask locus questions about the rectangular hyperbola.

 

[karnbmx]

my tip would be go back to basics, and write down step by step what you know you have to do.
Also, in the HSC exam, they can only ask locus questions about the rectangular hyperbola.

If that is the case, why is it part of questions in many 4Unit textbooks?

 

[deswa1]

Do you have any sample questions that you are having difficulty with plus the stages specifically where you have trouble? This would let us see what you understand and don't, plus you'll benefit from seeing a worked, explained solution.

 

[karnbmx]

Do you have any sample questions that you are having difficulty with plus the stages specifically where you have trouble? This would let us see what you understand and don't, plus you'll benefit from seeing a worked, explained solution.

Yeah, here is one:

The point P(acos(theta), bsin(theta)) is any point on the ellipse x^2/a^2 + y^2/b^2 = 1 with focus S.

The point M is the midpoint of interval SP.

Show that as P moves along the ellipse, M lies on another ellipse whose centre is midway between the origin O and the focus S.

Sorry. I can't use LaTeX very well, so I hope you don't mind the poor formatting. :S

 

[deswa1]

Allright. First off, draw a diagram and use parametric form because these questions are generally easier with it. I'll type up a worked solution- give me a few minutes.

<a href="http://www.codecogs.com/eqnedit.php?latex=\textup{Note the three points }P(acos\theta,bsin\theta), S(ae,0), M(\frac{a(cos\theta@plus;e)}{2},\frac{bsin\theta}{2})\\ \textup{We want to eliminate the theta's from the M value}\\ x=\frac{a(cos\theta@plus;e)}{2}, y=\frac{bsin\theta}{2}\\ \textup{Remember that }sin^2\theta@plus;cos^2\theta=1 \textup{ and rearranging the above to use this: }\\ cos\theta=\frac{2x-ae}{a}, sin\theta=\frac{2y}{b}\\ \frac{(2x-ae)^2)}{a^2}@plus;\frac{4y^2}{b^2}=sin^2\theta@plus;cos^2\theta=1\\ \textup{which is an ellipse centre }\frac{ae}{2},0" target="_blank"><img src="http://latex.codecogs.com/gif.latex?\textup{Note the three points }P(acos\theta,bsin\theta), S(ae,0), M(\frac{a(cos\theta+e)}{2},\frac{bsin\theta}{2})\\ \textup{We want to eliminate the theta's from the M value}\\ x=\frac{a(cos\theta+e)}{2}, y=\frac{bsin\theta}{2}\\ \textup{Remember that }sin^2\theta+cos^2\theta=1 \textup{ and rearranging the above to use this: }\\ cos\theta=\frac{2x-ae}{a}, sin\theta=\frac{2y}{b}\\ \frac{(2x-ae)^2)}{a^2}+\frac{4y^2}{b^2}=sin^2\theta+cos^2\theta=1\\ \textup{which is an ellipse centre }\frac{ae}{2},0" title="\textup{Note the three points }P(acos\theta,bsin\theta), S(ae,0), M(\frac{a(cos\theta+e)}{2},\frac{bsin\theta}{2})\\ \textup{We want to eliminate the theta's from the M value}\\ x=\frac{a(cos\theta+e)}{2}, y=\frac{bsin\theta}{2}\\ \textup{Remember that }sin^2\theta+cos^2\theta=1 \textup{ and rearranging the above to use this: }\\ cos\theta=\frac{2x-ae}{a}, sin\theta=\frac{2y}{b}\\ \frac{(2x-ae)^2)}{a^2}+\frac{4y^2}{b^2}=sin^2\theta+cos^2\theta=1\\ \textup{which is an ellipse centre }\frac{ae}{2},0" /></a>

 

[karnbmx]

Allright. First off, draw a diagram and use parametric form because these questions are generally easier with it. I'll type up a worked solution- give me a few minutes.

*facepalm* that looks so easy!!!!!!!!!!!!!!!!

 

[deswa1]

*facepalm* that looks so easy!!!!!!!!!!!!!!!!

Yeah. It's not really that hard. Basically you just need to look at the x and y values and look to see how you can eliminate the parameters.