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A circle is tangent to both the x and y-axis and the equation x+y=8. what are the equations of the circle?
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The formulation is, OBVIOUSLY, not exactly correct and, therefore, is not pure Math.
The correct formulation, simultaneously from the Math and the Grammar point of view, is THIS:
A circle is tangent to both the x and y-axis and to the straight line with the equation x+y=8. Find the circles.
Solution
Since the circle is tangent to both the x and y-axis, its center lies at the bisector of the first quadrant angle x = y, and r = x = y.
Now we have two cases (see the Figure below).
Case 1. Small circle inside the triangle.
Coordinates of the center via the radius x = y = r.
Coordinates of the tangent point via the radius = = .
Since + = 8, it gives an equation for "r"
+ + + = 8,
= 4 ====> r = = 2.34 (approx.)
Plot x + y = 8 and two circles.
Case 2. Large circle outside the triangle.
Coordinates of the center via the radius x = y = R.
Coordinates of the tangent point via the radius = = .
Since + = 8, it gives an equation for "R"
- + - = 8,
= 4 ====> R = = 13.66 (approx.)
Answer. Small circle radius r = = 2.34 (approx.) and the center x = y = r. The equation is + = .
Large circle radius R = = 13.66 (approx.) and the center x = y = R. The equation is + = .
Solved.