Find the equation of the circle which passes through the points (2,−2), and (3,4) and whose centre lies on the line x+y=2.
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Step-by-Step Solution
Step 1: General Equation of a Circle
The general equation of a circle with center (h,k) and radius r is given by (x−h)2+(y−k)2=r2. We need to find the values of h, k, and r to determine the equation of the circle.
Step 2: Formulate Equations from Given Points
Since the circle passes through the points (2,−2) and (3,4), these points must satisfy the circle's equation. Substituting these coordinates into the general equation gives us two equations involving h, k, and r.
Step 3: Equate the two expressions for r squared
Since both expressions are equal to r2, we can set them equal to each other. This eliminates r and gives us an equation solely in terms of h and k. Expanding and simplifying this equation will help us find a relationship between h and k.
Step 4: Simplify the equation for h and k
Expanding the squared terms and simplifying the equation from the previous step, we get a linear equation relating h and k. This equation, 2h+12k=17, represents the perpendicular bisector of the segment connecting the two given points.
Step 5: Use the condition that the center lies on the line
We are given that the center (h,k) lies on the line x+y=2. Substituting h for x and k for y gives us another linear equation: h+k=2. Now we have a system of two linear equations with two variables, h and k.
Step 6: Solve the system of linear equations
We solve the system of equations: 2h+12k=17 and h+k=2. From the second equation, we express h as 2−k and substitute it into the first equation. This allows us to find k=1013, and then h=107. So, the center of the circle is (107,1013).
Step 7: Calculate the radius and final equation
Now that we have the center (h,k)=(107,1013), we can find r2 by substituting these values and one of the given points (e.g., (2,−2)) into the circle equation. After calculating r2=50629, we write the final equation of the circle.