(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

The points [tex]P(2ap, ap^{2})[/tex] and [tex]Q(2aq, aq^{2})[/tex] lie on the parabola [tex]x^{2} = 4ay[/tex].

The equation of the normal to the parabola at P is [tex]x + py = 2ap + ap^{3}[/tex] and

the equation of the normal at Q is [tex]x + qy = 2aq + aq^{3}[/tex]. These normals intersect at R. Find the locus of R if PQ is a focal chord.

2. Relevant equations

equation of line:[tex]\frac{y-y_{1}}{x-x_{1}}=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}[/tex]

[tex]\frac{dy}{dx}=p=q[/tex], p & q are parameters on the parabola [tex]x^{2}=4ay[/tex]

focal chord (chord passing through focus [0,a]) is given by: [tex]y=\frac{1}{2}(p+q)x-apq[/tex]

if focal chord, pq=-1

3. The attempt at a solution

The coordinates of R is [tex] ( –apq[p + q] , a[p^{2}+pq+q^{2}+2] ) [/tex] and since PQ is a focal chord, pq=-1, therefore this simplifies to [tex] ( -a[p+q] , a[p^{2}+q^{2}+1] ) [/tex]

From here I am completely stumped on what I need to do. I'm even unsure if the coordinates of R is necessary in this question.

Any ideas of suggestions for how I can begin to approach this question?

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# Homework Help: Parametric parabolic loci

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