Conics Locus Question (1 Viewer)

Ambility · Apr 3, 2016

If someone could do a worked solution for part ii that would be great. I have already done part i.

$The line through O the origin is perpendicular tot he tangent at $P(cp,\frac{c}{p})$ on the rectangular hyperbola $xy=c^2$ meets the tangent at N. You may assume this tangent has the equation $x+p^2y=2cp$.$\\\\$i. Show that the coordinates of N are $(\frac{2cp}{1+p^4},\frac{2cp^3}{1+p^4})\\$ii. Hence or otherwise find the locus of N as p varies.$

Paradoxica · Apr 3, 2016

Ambility said:
If someone could do a worked solution for part ii that would be great. I have already done part i.

$The line through O the origin is perpendicular tot he tangent at $P(cp,\frac{c}{p})$ on the rectangular hyperbola $xy=c^2$ meets the tangent at N. You may assume this tangent has the equation $x+p^2y=2cp$.$\\\\$i. Show that the coordinates of N are $(\frac{2cp}{1+p^4},\frac{2cp^3}{1+p^4})\\$ii. Hence or otherwise find the locus of N as p varies.$

Consider the ratio of the co-ordinates of N, and their product.

Ambility · Apr 3, 2016

Paradoxica said:
Consider the ratio of the co-ordinates of N, and their product.

Thank you, but I'm going to need a little bit more than that. I take it I need to eliminate p, how can I do that?

KingOfActing · Apr 3, 2016

Ambility said:
If someone could do a worked solution for part ii that would be great. I have already done part i.

$The line through O the origin is perpendicular tot he tangent at $P(cp,\frac{c}{p})$ on the rectangular hyperbola $xy=c^2$ meets the tangent at N. You may assume this tangent has the equation $x+p^2y=2cp$.$\\\\$i. Show that the coordinates of N are $(\frac{2cp}{1+p^4},\frac{2cp^3}{1+p^4})\\$ii. Hence or otherwise find the locus of N as p varies.$

~~Are you sure those are the right coordinates of N? Not even wolfram alpha can find a Cartesian form for that locus (and it's plot isn't a shape that I recognise, at least.).~~

Edit: I take that back, thanks Paradoxica

Paradoxica · Apr 3, 2016

Ambility said:
Thank you, but I'm going to need a little bit more than that. I take it I need to eliminate p, how can I do that?

you cannot eliminate p itself, but you can eliminate p^2. Like I said, consider the quotient of the ordinates.

Conics Locus Question (1 Viewer)

Ambility

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Paradoxica

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Ambility

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KingOfActing

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Paradoxica

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