Fitzpatrick 3u Question Parametrics (1 Viewer)

silentk99 · Dec 20, 2015

Im stuck on this question for 4 hours. Its from Fitzpatrick 3u ex 24c, q 24. Any help appreciated.

P(2at,at^2) and q(2as,as^2) are two points on the parabola x^2=4ay. The chord pq subtends an right angle at origin O. If r is the fourth vertex of the rectangle poqr, show that:

1)ST=-4 (i GOT THIS PART)

2) Show that R is represented parametrically by x = 2a(t^2-4)/t, y=a(t^4+16)/t^2 (Need help with this part.)

leehuan · Dec 20, 2015

This is the wrong thread to be posting this >_>

Also, what is T?

b0b101 · Dec 20, 2015

silentk99 said:
Im stuck on this question for 4 hours. Its from Fitzpatrick 3u ex 24c, q 24. Any help appreciated.

P(2ap,ap^2) and q(2as,as^2) are two points on the parabola x^2=4ay. The chord pq subtends an right angle at origin O. If r is the fourth vertex of the rectangle poqr, show that:

1)ST=-4 (i GOT THIS PART)

2) Show that R is represented parametrically by x = 2a(t^2-4)/t, y=a(t^4+16)/t^2 (Need help with this part.)

$P$ (2ap,ap^2)$ and q$(2as,as^2)$ are two points on the parabola$ x^2=4ay. $The chord PQ subtends a right angle at origin O. If r is the fourth vertex of the rectangle POQR, show that:$\\1)ST=-4 $(I GOT THIS PART)$\\2) $Show that R is represented parametrically by $ x = \frac{2a(t^2-4)}{t}, y= \frac{a(t^4+16)}{t^2} $(Need help with this part.)$
Made it nice and neat for you...

silentk99 · Dec 20, 2015

b0b101 said:
$P$ (2ap,ap^2)$ and q$(2as,as^2)$ are two points on the parabola$ x^2=4ay. $The chord PQ subtends a right angle at origin O. If r is the fourth vertex of the rectangle POQR, show that:$\\1)ST=-4 $(I GOT THIS PART)$\\2) $Show that R is represented parametrically by $ x = \frac{2a(t^2-4)}{t}, y= \frac{a(t^4+16)}{t^2} $(Need help with this part.)$
Made it nice and neat for you...

Thanks,

Its not p but t

InteGrand · Dec 20, 2015

silentk99 said:
Im stuck on this question for 4 hours. Its from Fitzpatrick 3u ex 24c, q 24. Any help appreciated.

P(2at,at^2) and q(2as,as^2) are two points on the parabola x^2=4ay. The chord pq subtends an right angle at origin O. If r is the fourth vertex of the rectangle poqr, show that:

1)ST=-4 (i GOT THIS PART)

2) Show that R is represented parametrically by x = 2a(t^2-4)/t, y=a(t^4+16)/t^2 (Need help with this part.)

$\noindent One way to do it would be to use the fact that $R$ is the point of intersection of the line through $P$ parallel to $OQ$ and the line through $Q$ parallel to $OP$.$

silentk99 · Dec 21, 2015

InteGrand said:
$\noindent One way to do it would be to use the fact that $R$ is the point of intersection of the line through $P$ parallel to $OQ$ and the line through $Q$ parallel to $OP$.$

I tried that but the algebra was not working

Ambility · Dec 21, 2015

silentk99 said:
I tried that but the algebra was not working

Here's the full algebra working for you then. This is done using the method Integrand suggested, finding the equations for QR and PR, and then finding R by simultaneously solving these equations.
$\\s=\frac{-4}{t}$ (1)$\\m(QR)=m(OP)=\frac{t}{2}$ (sides of a rectangle)$\\$eqn of QR: $y-as^2=\frac{t}{2}(x-2as)\\2y-2as^2=tx-2ast$ (2)$\\$eqn of PR: $y-at^2=\frac{s}{2}(x-2at)\\2y-2at^2=sx-2ast$ (3)$\\$(2)-(3): $-2as^2+2at^2=tx-sx\\x(t-s)=2a(t+s)(t-s)\\x=2a(t+s)\\$Subing in (1): $x=2a(t+\frac{-4}{t})\\x=2a(\frac{t^2-4}{t})$ (4)$

$\\$Subing (4) into (2): $2y-2as^2=2at^2-8a-2ast\\2y=a(2t^2+2s^2-2st-8)\\$Using (1): $2y=a(2t^2+\frac{32}{t^2}-2\frac{-4}{t}t-8)\\2y=a(2t^2+\frac{32}{t^2})\\y=a(\frac{t^4}{t^2}+\frac{16}{t^2})=a(\frac{t^4+16}{t^2})$

Fitzpatrick 3u Question Parametrics (1 Viewer)

silentk99

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