Explanation needed (1 Viewer)

[Viscass]

Hey, there was an example in the Terry Lee text book in the rectangular hyperbola section which i generally get, just one thing i dont understand,

here is the question

Example 5.8.
PQ is a chord of the rectangular hyperbola xy=c^2. if PQ has a constant length of k, find the locus of R, the midpoint of PQ

the final line was,
.'. the locus of R is 4(x*y-c^2)(x^2+y^2)=k^2*x*y
and I'm kinda confused, because if you sub c^2=xy you get,
k^2*x*y =0
which would mean that either k=0 (the distance does not exist, point P = point Q) or x=0 / y=0 (which lie on the asymptote and k would be undefined)
could someone please explain this to me

thanks

 

[funnytomato]

xy=c^2 only holds on the hyperbola

the locus found is for R, the midpoint of PQ, which does not necessarily lie on the hyperbola
so in general you cannot assume xy=c^2 on this locus

dunno if that's clear enough, you could try to sketch it and see what happens

 

[Viscass]

xy=c^2 only holds on the hyperbola

the locus found is for R, the midpoint of PQ, which does not necessarily lie on the hyperbola
so in general you cannot assume xy=c^2 on this locus

dunno if that's clear enough

ah okay, understood thankyou so much
ill inform my teacher because he was also stumped when i asked him

 

[hayabusaboston]

xy=c^2 only holds on the hyperbola

the locus found is for R, the midpoint of PQ, which does not necessarily lie on the hyperbola
so in general you cannot assume xy=c^2 on this locus

dunno if that's clear enough, you could try to sketch it and see what happens

This