2006 iTest Problems/Problem U6
The following problem is from the Ultimate Question of the 2006 iTest, where solving this problem required the answer of a previous problem. When the problem is rewritten, the T-value is substituted.
Problem
and
are nonzero real numbers such that
The smallest possible value of is equal to
where
and
are relatively prime positive integers. Find
.
Solutions
Solution 1
Notice that and
. This suggests that we can factor the entire expression.
By rearranging terms, we have
By the Zero Product Property, either
or
. In the first case,
. In the second case, the equation can be rearranged into
, a circle with center
and radius
.
The values of and
must be on the circle for the second case. In addition, the line connecting
and the origin has a slope of
. Therefore, we need to find the smallest slope of a line connecting a point on the circle to the origin. That line must be tangent to the circle.
By drawing tangent lines and letting
be half the smaller angle formed between the two tangent lines (as seen in diagram), we find that
. Using the Double Angle Identity yields
, so
.
Since the slope of the line tangent to the circle is , the smallest value of
is
, so
.
Solution 2 (credit to NikoIsLife:))
Let , and we want
to be minimized. Substituting for
results in
Since
, we must have
or
. In the latter case,
The minimum value of
is
, so
.
See Also
2006 iTest (Problems, Answer Key) | ||
Preceded by: Problem U5 |
Followed by: Problem U7 | |
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