Introduction

This article discusses the method of finding the equation of the largest circle that lies between two parallel lines and whose center lies on a given line. Specifically, we explore the problem where the two lines are x - y 1 and x - y -3, and the center of the circle lies on the line (2xy 3).

Step-by-Step Solution

The first step in solving this problem is to calculate the distance between the two parallel lines, which will be the diameter of the circle. The distance (d) between two parallel lines (ax by c_1 0) and (ax by c_2 0) is given by the formula:

[ d frac{|c_1 - c_2|}{sqrt{a^2 b^2}} ]

For our lines, (a 1), (b -1), and the constants are (c_1 1) and (c_2 3).

[ d frac{|1 - 3|}{sqrt{1^2 (-1)^2}} frac{2}{sqrt{2}} sqrt{2} cdot frac{2}{sqrt{2}} 2sqrt{2} ]

The diameter of the circle is (2sqrt{2}), so the radius is (sqrt{2}).

Step 2: Determine the Center of the Circle

The center of the circle must lie on the line equidistant from both parallel lines. The midpoint of the perpendicular distance between the lines is the line (x - y -1). To find the exact coordinates of the center, we solve the system of equations:

The center lies on the line:

[ 2xy 3 ]

And the center also lies on the line:

[ x - y -1 ]

Solving these equations, we get:

From the second equation, we can express (y) in terms of (x): [ y x 1 ]

Substituting this into the first equation:

[ 2x(x 1) 3 ] [ 2x^2 2x - 3 0 ]

Solving this quadratic equation:

[ x frac{-2 pm sqrt{4 24}}{4} frac{-2 pm sqrt{28}}{4} frac{-2 pm 2sqrt{7}}{4} frac{-1 pm sqrt{7}}{2} ]

The quadrant constraints suggest that we take the midpoint solution, thus:

[ x frac{2}{3}, y frac{5}{3} ]

[ C left(frac{2}{3}, frac{5}{3}right) ]

Step 3: Formulate the Equation of the Circle

The general equation of a circle with center ((h, k)) and radius (r) is:

[ (x - h)^2 (y - k)^2 r^2 ]

Substituting (h frac{2}{3}), (k frac{5}{3}), and (r sqrt{2}):

[ left(x - frac{2}{3}right)^2 left(y - frac{5}{3}right)^2 2 ]

Conclusion

The equation of the largest circle that lies between the lines (x - y 1) and (x - y -3) with the center on the line (2xy 3) is:

[ boxed{left(x - frac{2}{3}right)^2 left(y - frac{5}{3}right)^2 2} ]