another conics locus problem (1 Viewer)

[freaking_out]

a variable chord of the rectangular hyperbola xy=c² is such that its projection on the x-axis has constant length 2c, show that the locus of its mid-point has equation X²Y=c²(X+Y).

let the chord have extremeties P(x₁,y₁), Q(x₂,y₂) and the midpoint be M(X,Y). (note that 2c=x₁-x₂)

 

[Affinity]

well....

let's call x(1) p instead.

then x(2) = p+2c

solving we get
y(1) = c^2/p
y(2) = c^2/(p+2c)

the locus of the mid point has parametric form (p as parameter)

x = p+c
y = [c^2/p + c^2/(p+2c)]/2

notice
p + 2c = x+c
p = x- c
so eliminating p

y = [c^2/(x-c) + c^2/(x+c)]/2

y = (c^2/2)(2x/(x^2-c^2))

y = xc^2/(x^2-c^2)

yx^2 - yc^2 = xc^2

(x^2)y = c^2(x+y)

as required

 

[freaking_out]

gosh, u people r so smart at conics! :worried: