Parametric help (1 Viewer)

[Footyking]

Q1: If the points P(8p,4p^2) and Q(8q,4q^2) lie on the parabola x^2=16y and R [4(p+q), 4pq] is the point in which the tangents at p and q meet. What is the Cartesian equation for the locus of R?

Q2: The equation of the chord of PQ is y= (p+q/2)x -apq. M is the midpoint of the focal chord PQ which is (ap+aq, ap^2+aq^2/2) and N is a point on the directric such that MN is perpendicular to the directrix. T is the midpoint of MN which is [a(p+q), a(p^2+q^2-2)/4]. Find the equation of the locus of T.

Q3: The chord of contact of the tangents to the parabola x^2 = 4ay from an external point A(x1,y1) passes through the point B (0,2a). Find the equation of the locus of the midpoint AB

IF anybody can help on any of these questions it would be greatly appreciated.

 

[braintic]

In question 1, there needs to be more information. Such as P and Q are the endpoints of a focal chord. Or OP is perpendicular to OQ. Some information to link P and Q.

Q3: CofC: xx1=2a(y+y1)
Passes thru (0,2a): (0)(x1)=2a(0+y1)
y1=0
So CofC is xx1=2ay
ie. y=xx1/2a
Solve simult with eqn of parabola: x^2 = 4a(xx1/2a)
x^2 = 2xx1
x(x-2x1)=0
x=0, 2x1
y=0, x1^2/2a
Midpoint: (x1, x1^2/4a)
ie. y=x^2/4a
x^2 = 4ay