power of conic
Narasimham
origin with product of its lengths equalling 1.(Similar to power of
co-axal circles). What is its equation ?
Robert Israel
You have removed only one degree of freedom with your condition.
I assume that by "its lengths" you mean the lengths of the line
segments from the origin to the points of tangency. Take any conic
that has two tangents drawn from the origin, and scale appropriately
to get an example.
Robert Israel isr...@math.ubc.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada
Narasimham
Rephrasing the same question as:
Find equation of all conics including arbitrary constant/costants so
that the product of tangent length segments from origin to point of
tangency always equals 1.
x^2 - 2 x h + y^2 + 1 = 0 is a simple example of a circle of variable
radius, h being the arbitrary constant representing distance between
origin and center of circle on x-axis.Its conical generalization is in
fact asked for.
I suspect that it may be the real/imaginary part of an unknown (to me)
function of a complex variable with two arbitrary constants.
Narasimham
> Rephrasing the same question as:
>
> Find equation of all conics including arbitrary constant/costants so
> that the product of tangent length segments from origin to point of
> tangency always equals 1.
>
> x^2 - 2 x h + y^2 + 1 = 0 is a simple example of a circle of variable
> radius, h being the arbitrary constant representing distance between
> origin and center of circle on x-axis.Its conical generalization is in
> fact asked for.
Narasimham wrote to
> rephrase the same question as:
>
> Find equation of all conics including arbitrary constant/costants so
> that the product of tangent length segments from origin to point of
> tangency always equals 1.
>
> x^2 - 2 x h + y^2 + 1 = 0 is a simple example of a circle of variable
> radius, h being the arbitrary constant representing distance between
> origin and center of circle on x-axis.Its conical generalization is in
> fact asked for.
Implicit plots of several conics a x^2 + 2 h x y + b y^2 + 2 f x + 2 g
y + 1 = 0 appear to satisfy this from the origin as an "external"
point,so posted the previous "laborious old conics question". The
generalization for internal points is not clear.May be it is given in
one of those classical Chelsea Publications on conics that I cannot
access now.Also comments are needed to reconcile both these questions.