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Find the equation of a circle that is circumscribes a triangle determined by the lines y = 0, y = x, and 2x + 3y = 10.
Solution:
To illustrate the problem, it is better to draw the figure as follows
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| A circle that is circumscribes a triangle from the intersection of three lines. (Photo by Math Principles in Everyday Life) |
The first thing that we need to do is to get the intersections of three lines that passes the curve of a circle.
To get the coordinates of A, use equations y = 0 and y = x as follows:
Since the intersection of y = 0 and y = x is (0, 0) as shown from the figure, then the first point of intersection is A (0, 0).
To get the coordinates of B, use equations y = x and 2x + 3y = 10 as follows:
but
then the above equation becomes,
If x = 2, then y = 2. Hence, the second point of intersection is B (2, 2).
To get the coordinates of C, use equations y = 0 and 2x + 3y = 10 as follows:
but
then the above equation becomes,
Hence, the third point of intersection is C (5, 0).
The general equation of a circle is
Since the three points of a circle are given, then we can solve for the values of D, E, and F.
If you will use A (0, 0), then the above equation becomes
If you will use C (5, 0), then the above equation becomes
but F = 0, then the above equation becomes
If you will use B (2, 2), then the above equation becomes
Substitute D = - 5 and F = 0, we have
Therefore, the equation of a circle is
