
Analytic Geometry
Hello, everybody. I need help... (Worried)
Show that, for all values of p, the point P given by x=ap^{2}, y=2ap lies on the curve y^{2}=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 (aq^{2}, 2aq). Show that p^{2}+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 y^{2}(x+2a) +4a^{3}=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 y^{2}(x+2a) +4a^{3}=0” in Q (b). Can somebody help me?
Thanks!!(Happy)

Re: Analytic Geometry
Presumably, you have found that that $\displaystyle R$ has the coordinates:
$\displaystyle (apq,a(q+p))$
Now, use the relationship between $\displaystyle p$ and $\displaystyle q$ you found earlier:
$\displaystyle p^2+pq+2=0$
which when solved for $\displaystyle q$ is:
$\displaystyle q=\frac{p^2+2}{p}$
Now you may write the coordinates of $\displaystyle R$ as parametric equations in one parameter, which you may then eliminate to obtain the required Cartesian equation.

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.