Analytic Geometry

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Sep 28th 2012, 08:16 PM
AuXian

Analytic Geometry

Hello, everybody. I need help... (Worried)
Show that, for all values of p, the point P given by x=ap², y=2ap lies on the curve y²=4ax.
a) Find the equation of this normal to this curve at the point P.
If this normal meets the curve again at the point Q (aq², 2aq). Show that p²+pq+2=0
b) Determine the coordinates of R, the point of intersection of the tangents of the curve at the point P and Q.
Hence, show that the locus of the point R is y²(x+2a) +4a³=0

I already solved Q (a) and first question of Q (b). However, I can’t solve the last question: “Hence, show that the locus of the point R is y²(x+2a) +4a³=0” in Q (b). Can somebody help me?
Thanks!!(Happy)
Sep 28th 2012, 08:29 PM
MarkFL

Re: Analytic Geometry

Presumably, you have found that that $R$ has the coordinates:

$(apq,a(q+p))$

Now, use the relationship between $p$ and $q$ you found earlier:

$p^2+pq+2=0$

which when solved for $q$ is:

$q=-\frac{p^2+2}{p}$

Now you may write the coordinates of $R$ as parametric equations in one parameter, which you may then eliminate to obtain the required Cartesian equation.
Sep 28th 2012, 08:34 PM
chiro

Re: Analytic Geometry

Hey AuXian.

I haven't seen this stuff since high school (more than 10 years) but the wiki page gives a derivation of the locus for the parabola:

Parabola - Wikipedia, the free encyclopedia

I'm not sure if you just formulas or have to derive things, but if you just formulas the wiki gives a derivation from start to finish in terms of the a term.