Difference between revisions of "1976 USAMO Problems/Problem 2"
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\end{align*} </cmath> | \end{align*} </cmath> | ||
Solving these simultaneous equations gives coordinates for <math>P</math> in terms of <math>a, b, r,</math> and <math>s</math>: | Solving these simultaneous equations gives coordinates for <math>P</math> in terms of <math>a, b, r,</math> and <math>s</math>: | ||
− | <math>P = (\frac{as}{b},1-ar)</math>. These coordinates can be parametrized as follows: | + | <math>P = \left(\frac{as}{b},\frac{1 - ar}{b}\right)</math>. These coordinates can be parametrized in Cartesian variables as follows: |
<cmath> \begin{align*} | <cmath> \begin{align*} | ||
− | + | x &= \frac{as}{b}\\ | |
− | + | y &= \frac{1 - ar}{b}. | |
\end{align*} </cmath> | \end{align*} </cmath> | ||
− | Now | + | Now solving for <math>r</math> and <math>s</math> to get <math>r = \frac{1-by}{a}</math> and <math>s = \frac{bx}{a}</math> . Then since <math>r^s + s^2 = 1, \left(\frac{bx}{a}\right)^2 + \left(\frac{1-by}{a}\right)^2 = 1</math> which reduces to <math>x^2 + (y-1/b)^2 = \frac{a^2}{b^2}.</math> This equation defines a circle and is the locus of intersection points <math>P</math>. |
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==See also== | ==See also== |
Revision as of 17:19, 22 February 2012
Problem
If and
are fixed points on a given circle and
is a variable diameter of the same circle, determine the locus of the point of intersection of lines
and
. You may assume that
is not a diameter.
Solution
WLOG, assume that the circle is the unit circle centered at the origin. Then the points and
have coordinates
and
respectively and
and
have coordinates
and
. Then we can find equations for the lines:
Solving these simultaneous equations gives coordinates for
in terms of
and
:
. These coordinates can be parametrized in Cartesian variables as follows:
Now solving for
and
to get
and
. Then since
which reduces to
This equation defines a circle and is the locus of intersection points
.
See also
1976 USAMO (Problems • Resources) | ||
Preceded by Problem 1 |
Followed by Problem 3 | |
1 • 2 • 3 • 4 • 5 | ||
All USAMO Problems and Solutions |