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.)
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.
