parabola questions (1 Viewer)

[onebytwo]

QUESTION : on x^2=4ay, tangents at points P and Q intersect at S. given that p-q=1+pq, find the eqtn of the locus of S.
ans: (y+a)^2=x^2 - 4ay

QUESTION: on x^2=4ay, normals at P and Q intersect at N. given that chord PQ passes through (0,2a), find the locus of N
ans:x^2=4a(y-4a)

 

[SoulSearcher]

2. You can derive the equations of the normals and chord PQ for yourself, I'll just use the standard forms to make it easier to type.
With points P (2ap,ap²) and Q (2aq,aq²), the normals at those points would be
x+py=ap³+2ap ... (1)
x+qy=aq³+2aq ... (2)
Therefore the point of intersection is, by solving simutaneously (1) - (2),
py-qy=ap³-aq³+2ap-2aq
(p-q)y=a(p-q)(p²+pq+q²)+2a(p-q)
y = a[p²+pq+q²+2]
Therefore x is, substituting back into (1),
x+ap(p²+pq+q²+2)=ap³+2ap
x+apq(p+q)=0
x = -apq(p+q)
Equation of chord PQ is 2y = (p+q)x-2apq)
At point (0,2a)
4a = -2apq
pq = -2
Therefore, with
x = -apq(p+q), and
y = a[p²+pq+q²+2]
x = 2a(p+q)
y = a(p²+q²)
= a[(p+q)²-2pq]
=a[(p+q)²+4]
x/2a = p+q
.’. y = a(x²/4a²+4)
= x²/4a +4a
y-4a=x²/4a
4a(y-4a)=x²