Difference between revisions of "University of South Carolina High School Math Contest/1993 Exam/Problem 25"
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== Problem == | == Problem == | ||
+ | What is the center of the circle passing through the point <math>(6,0)</math> and tangent to the circle <math>x^2 + y^2 = 4</math> at <math>(0,2)</math>? (Two circles are tangent at a point <math>P</math> if they intersect at <math>P</math> and at no other point.) | ||
− | <center><math> \mathrm{(A) \ } \qquad \mathrm{(B) \ } \qquad \mathrm{(C) \ } \qquad \mathrm{(D) \ } \qquad \mathrm{(E) \ } | + | <center><math> \mathrm{(A) \ }(0,-6) \qquad \mathrm{(B) \ } (1,-9) \qquad \mathrm{(C) \ } (-1,-9) \qquad \mathrm{(D) \ } (0,-9) \qquad \mathrm{(E) \ } \rm{none \ } \rm{of \ } \rm{these} </math></center> |
== Solution == | == Solution == | ||
+ | Let the circle we are looking for be <math>(x-h)^{2}+(y-k)^{2}=r^{2}</math> where <math>(h,k)</math> is obviously the center. Plugging in points <math>(6,0)</math> and <math>(0,2)</math> gives us that <math>3k-h=8</math>. Seeing our answer choices, none of the points work, thus our answer is E. | ||
== See also == | == See also == | ||
* [[University of South Carolina High School Math Contest/1993 Exam]] | * [[University of South Carolina High School Math Contest/1993 Exam]] |
Revision as of 19:37, 22 July 2006
Problem
What is the center of the circle passing through the point and tangent to the circle
at
? (Two circles are tangent at a point
if they intersect at
and at no other point.)
![$\mathrm{(A) \ }(0,-6) \qquad \mathrm{(B) \ } (1,-9) \qquad \mathrm{(C) \ } (-1,-9) \qquad \mathrm{(D) \ } (0,-9) \qquad \mathrm{(E) \ } \rm{none \ } \rm{of \ } \rm{these}$](http://latex.artofproblemsolving.com/0/0/b/00bc059af7b25de3ab7957a6c887569ef06b1fdd.png)
Solution
Let the circle we are looking for be where
is obviously the center. Plugging in points
and
gives us that
. Seeing our answer choices, none of the points work, thus our answer is E.