Find the coordinated of a point.

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SUMMARY

The discussion focuses on finding the coordinates of point Q, which divides the line segment AP in the ratio of a²:b², where b > a. The equation derived for line PA is y - b tan(θ) = (b sec(θ) / (a sec(θ)))(x - a sec(θ)). The participants emphasize the importance of understanding the relationship between the segments AQ and QP in relation to the total length of AP. Additionally, the context involves hyperbolic geometry, as points A and B are identified as vertices of a hyperbola.

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Homework Statement



Find the coordinates of point Q if Q divides the line AP in the ratio a^{2}:b^{2}, b>a.

Homework Equations



http://i.imgur.com/g45qxl6.png

The Attempt at a Solution



I found the equation of PA which was y-btanθ=(bsecθ)/(asecθ)(x-asecθ) than I didn't know what to do.
This is supposed to be a very easy question but I'm not good at conics, I'm much better at calculus/analysis :)
 
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To say that Q is such that it divides the interval, AP, in the ratio x:y means that we must have |AQ|/|AP|= x/(x+y) and, as a result, |QP|/|AP|= y/(x+y).

Here, |AB| is the length of the interval so that we must have |AQ|+ |QP|= |AP|.

From your picture, as well as your refernce to "conics", it appears that A and B are the vertices of an hyperbola and P a point on that hyperbola but you don't seem to have used that fact anywhere.
 
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