Help on easy paramertic equation..please (1 Viewer)

[Gangstar1]

the normal at any point P(-8p,-4p^2) on the parabola x^2=-16y cuts the y-axis at point M. Find the equation of the locus of the midpoint PM

thank you

 

[xV1P3R]

Just curious, if it's easy then what's the problem?

 

[Gangstar1]

Find the equation of the locus of the midpoint PM, i need someone to show me the working out everyone in my class got it and i still cant get

 

[xV1P3R]

Just skipping the boring algebra, you agree that the equation of the normal at P is:
x + py = -4p(p²+1)

P(-8p,-4p²)
M (0 , -4(p²+1))

mid(PM) = (-4p , -4(2p² + 1))

Let L(x,y) be the general point on the required locus as P varies:
x = -4p
y = -4(2p² + 1)

p = -x/4
sub into u
y = -4(2(x²/16) + 1)
y = -8(x²/16) + 4
y - 4 = -x²/2
x² = -2(y-4)
which is your locus

 

[Gangstar1]

thank you