Parametrics Q (1 Viewer)

[mtsmahia]

Q) P is a variable point on the parabola x^2=-4y . The tangent of P cuts the parabola x^2=4y @ Q and R. Show that 3x^2=4y is the eq of the locus of the midpoint of the chord RQ.

thanks!

 

[martinc]

Q) P is a variable point on the parabola x^2=-4y . The tangent of P cuts the parabola x^2=4y @ Q and R. Show that 3x^2=4y is the eq of the locus of the midpoint of the chord RQ.

P is point (-2p,-p^2) since a=1

dy/dx = dy/dt / dxdt
dy/dx = -2p/-2
=p

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

cuts the parabola x^2=4y.. then solve simultaneously

y = x^2 / 4
y=px+p^2

x^2=4px+4p^2
x^2-4px-4p^2=0
to find two points of intersection.. solve for x
quadratic equation..
delta = 16p^2-4(1)(-4p^2)
=32p^2
x= (4p plusminus root32p^2) / 2
x=(2p plusminus 2rootp^2)
y=______

just sub in y.. i'm too lazy to do it.. then find midpoint and then eliminate the vairable.
it looks pretty messy. can someone find an error in my working?

 

[mtsmahia]

Q) P is a variable point on the parabola x^2=-4y . The tangent of P cuts the parabola x^2=4y @ Q and R. Show that 3x^2=4y is the eq of the locus of the midpoint of the chord RQ.

P is point (-2p,-p^2) since a=1

dy/dx = dy/dt / dxdt
dy/dx = -2p/-2
=p

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

cuts the parabola x^2=4y.. then solve simultaneously

y = x^2 / 4
y=px+p^2

x^2=4px+4p^2
x^2-4px-4p^2=0
to find two points of intersection.. solve for x
quadratic equation..
delta = 16p^2-4(1)(-4p^2)
=32p^2
x= (4p plusminus root32p^2) / 2
x=(2p plusminus 2rootp^2)
y=______

just sub in y.. i'm too lazy to do it.. then find midpoint and then eliminate the vairable.
it looks pretty messy. can someone find an error in my working?

Yeah , i ended up doing the same steps - looked messy to me to , i thought i did it wrong

 

[bored of sc]

Q) P is a variable point on the parabola x^2=-4y . The tangent of P cuts the parabola x^2=4y @ Q and R. Show that 3x^2=4y is the eq of the locus of the midpoint of the chord RQ.

P is point (-2p,-p^2) since a=1

dy/dx = dy/dp / dx/dp
can someone find an error in my working?

It's a trivial error but it's an error nonetheless.