Parametrics (1 Viewer)

[acmilan]

Can someone help me with some parametric questions.

1. Find the co-ordinates of three points on the parabola x^2 = 4y such that the normals through these three points pass through the point (-12,15) [Answer: (-2,1) (-6,9) (8,16)]

2.How many normals pass through (0,ka), a point on the axis of the parabola x^2 = 4ay, for k>2? For k = 3, find where the normal meets the parabola again [Answer: 2; (+or-6a,9a)]

3.The chord of contact of the tangents to the parqabola x^2 = 4ay from the point P[x(0), y(0)] passes through the point Q(0,2a). Show that the locus of the mid-point of PQ is the X-axis

 

[Estel]

1...
The points required: (2t, t^2) [t1, t2, t3... but i can't do subscripts ]
Eq normal: x + ty = 2t + t^3
Substituting (-12,15)
-12 + 15t = 2t + t^3
t^3 - 13t + 12 = 0
By inspection, t=1.
Then (t-1)(t^2+t-12)=0
t = 1, -4, 3
Giving (2,1) (-8,16) and (6, 9)

hmm
Where have I gone wrong.

 

[Estel]

2...
Eq normal: x + ty = 2at + at^3
Passing (0,ka)
kat = 2at + at^3
at(t^2+(2-k))=0
for k>2, 3 solutions (t=0, t=+-rt(k-2)

Meeting the parabola again:
for k=3, t(t^2-1)=0
t=0, 1, -1
x + ty = 2at + at^3
x^2 = 4ay... y = x^2/4a
4x + 4y = 12a OR 4x-4y = -12a OR x=0, which doesn't meet the parabola again.
x^2 + 4ax -12a^2 = 0 OR x^2 - 4ax - 12a^2 = 0
x= -6a or 6a
And y = 9a^2/a
= 9a

 

[Estel]

3...
Chord of contact:
xx(0) =2a[y+y(0)]
Passing (0,2a)
Then 2a[2a+y(0)]=0, y(0)=-2a
Midpt PQ:
y = [y(0)+2a]/2 = 0
Hence the locus of the midpt of PQ is the X-axis.

 

[acmilan]

thanks