Parametrics questions (1 Viewer)

[Dora Explorer]

1. P is a variable point on the parabola x² = -4y. The tangent from P cuts the parabola x² = 4y at Q and R. Show that 3x² = 4y is the equation of the locus of the mid-point of the chord RQ.

2. Show that the equation of the locus of R [a(p-1/p), a/2(p² + 1/p²)] is x² = 2a(y-a).

 

[jet]

 

[life92]

for the parabola x^2 = -4y, a = 1
therefore
P = (2p, -p^2)

dy/dx = (dy/dt) / (dx/dt)
= -2p / 2
= -p

tangent at P
y + p^2 = -p (x-2p)
y = -px + 2p^2 - p^2
y = -px + p^2

through x^2 = 4y
x^2 / 4 = -px + p^2
x^2 + 4px - 4p^2 = 0
x = [-4p +- root(16p^2 + 16p^2) ] / 2
x = -2p +- 2p root2
x = 2p [ -1 +- root2 ]

x2 = 2p [ -1 + root2 ]
x3 = 2p [ -1 - root2 ]

now, x^2 = 4y
4y2 = 4p^2 [ 1 - 2root2 + 2]
y2 = p^2 [ 3 - 2root2 ]

that is, R = ( 2p[-1+root2] , p^2[3-2root2] )

for Q,
4y3 = 4p^2 [ 1 + 2root2 +2 ]
y3 = p^2 [ 3 + 2root2 ]

that is, Q = ( 2p[ -1 - root2] , p^2[3+2root2] )

now, midpoint of RQ = (-2p, 3p^2)
x = -2p
p = -x/2

y = 3p^2
y = 3 . (-x/2) ^2
y = 3x^2 / 4
4y = 3x^2

ahh..
hopefully you understand my working xS
sorry, still dno how to use latex ,,

 

[Dora Explorer]

hey thanks for that

and dw, i'll look into it - it makes sense