# 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 $\displaystyle -p$ and $\displaystyle -q$, so the normals are perpendicular gives $\displaystyle pq = -1$.

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