If the line `2x-y+1=0` touches the circle at the point `(2,5)` and the centre of the circle lies in the line `x+y-9=0.` Find the equation of the circle.

Explanation

To find the equation of the circle that touches the line \(2x - y + 1 = 0\) at the point \((2, 5)\) and has its center on the line \(x + y - 9 = 0\), we can follow these steps:

### Step 1: Identify the center of the circle
Since the center of the circle lies on the line \(x + y - 9 = 0\), we can express the coordinates of the center \((a, b)\) as:
\[ a + b = 9 \quad \text{(1)} \]
From this equation, we can express \(a\) in terms of \(b\):
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