# Thread: locus of parabola problems

1. ## locus of parabola problems

a) The normals to the parabola x^2 = 4ay at the points P(2ap, ap^2) and Q(2aq, aq^2) are perpendicular to each other and meet at R. Prove that R lies on the parabola x^2 = a(y-3a)

b) P is a variable point on the parabola x^2=4ay and N is the foot of the perpendicular drawn from the focus S to the normal at P. Show that the locus of N is x^2 = a(y-a)

if anyone could help me with these problems, i would REALLY appreciate it, thankYOU !

2. Originally Posted by flyinhigh123
a) The normals to the parabola x^2 = 4ay at the points P(2ap, ap^2) and Q(2aq, aq^2) are perpendicular to each other and meet at R. Prove that R lies on the parabola x^2 = a(y-3a)

b) P is a variable point on the parabola x^2=4ay and N is the foot of the perpendicular drawn from the focus S to the normal at P. Show that the locus of N is x^2 = a(y-a)

if anyone could help me with these problems, i would REALLY appreciate it, thankYOU !
The gradient of the normals at P and Q are $-p$ and $-q$, so the normals are perpendicular gives $pq = -1$.

Now use the technique I showed you in my posting to your previous question (where $pq = -4$). Use the same equations for $x$ and $y$ - the ones that represent the point of intersection of the normals; eliminate $pq$ in the equation for $x$; square both sides; then form an expression for $y$ in terms of $(p^2+q^2)$ and $pq$; then eliminate $p$ and $q$ altogether, and you're done.