SOLUTION: A hyperbolic mirror has the property that a light ray directed at a focus will be reflected to the other focus. The focus of a hyperbolic mirror has coordinates (-10,0). Find the v

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Question 1194562: A hyperbolic mirror has the property that a light ray directed at a focus will be reflected to the other focus. The focus of a hyperbolic mirror has coordinates (-10,0). Find the vertex of the mirror if the mounting point of the mirror is located at (10,10). Round your answer to the nearest hundredth.

Found 2 solutions by MathLover1, ikleyn:
Answer by MathLover1(20849) About Me  (Show Source):
You can put this solution on YOUR website!
given:
The focus of a hyperbolic mirror has coordinates (-10,0) => other focus is at (10,0)
the mounting point of the mirror is located at M=(10,10)

hyperbola equation:
%28x-h%29%5E2%2Fa%5E2-%28y-k%29%5E2%2Fb%5E2=1 where (h,k) is the center,+a and b are the lengths of the semi-major and the semi-minor axes

since center is half way between foci, (h,k)=(0,0)

then we have

%28x-0%29%5E2%2Fa%5E2-%28y-0%29%5E2%2Fb%5E2=1
x%5E2%2Fa%5E2-y%5E2%2Fb%5E2=1

since the focus is at (c,0)=(10,0)=> c=10+
since a%5E2+%2B+b%5E2+=+c%5E2
a%5E2+%2Bb%5E2=10%5E2
+a%5E2+%2Bb%5E2=100
+b%5E2+=100-a%5E2

then
x%5E2%2Fa%5E2-y%5E2%2F%28100-a%5E2%29=1 ........use point M=(10,10)
100%2Fa%5E2-100%2F%28100-a%5E2%29=1

solving this we get 4 solutions an one of them is:

a+=+5+%28sqrt%285%29+-+1%29 =>a%5E2=150+-50sqrt%285%29
then
b%5E2=100-%28150+-50sqrt%285%29%29=50sqrt%285%29+-+50

x^2/(150 -50sqrt(5))-y^2/(50sqrt(5) - 50)=1.......exact solution

round to the nearest hundredth:
a%5E2=38.19+

b%5E2=61.80

x%5E2%2F38.19+-y%5E2%2F61.80=1








Answer by ikleyn(52781) About Me  (Show Source):
You can put this solution on YOUR website!
.
A hyperbolic mirror has the property that a light ray directed at a focus will be reflected to the other focus.
The focus of a hyperbolic mirror has coordinates (-10,0). Find the vertex of the mirror if the mounting point
of the mirror is located at (10,10). Round your answer to the nearest hundredth.
~~~~~~~~~~~~~~~~~~~~


As worded,  printed,  posted and presented,  this problem is  INCOMPLETE  and,  therefore,  can not be solved.

It is incomplete,  because giving the position of one focus of hyperbola and one point on the hyperbola
is not enough to identify the hyperbola by a unique way.


So,  in addition to the given info, more information should be given  EITHER  related to a/the vertex of the hyperbola,
OR  to its other focus,  OR  to its center.


In her solution, @MathLover implicitly assumes that the other focus is symmetric  (10,0)
to the first focus and thus the center of the hyperbola is the origin of the coordinate system,
but formally she  HAS  NO  a solid base to make such an assumption.


I know that the visitors of this forum often re-post  (copy-paste)  the problems from other web-sites
without been competent in  Math problems formulations.

It is exactly that case,  when incomplete formulation roams/migrates from site to site,  littering the  Internet.


In addition,  the optical property of hyperbolas is formulated and presented  INCORRECTLY  in this post.

For the correct formulation of this property,  see the lesson
    - Optical property of a hyperbola.
in this site.