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Step-by-Step Solution
Step 1: Understand the Problem
We need a circle in the first quadrant whose center lies on the line
x + y = 2 , and which is tangent to the lines x = 3 and y = 2 . We must find the diameter of this circle.
Step 2: Represent the Center of the Circle
Let the center of the circle be at the point (h, k) . Since the center lies on x + y = 2 , it follows that
h + k = 2 \quad \Rightarrow \quad k = 2 - h.
Step 3: Express the Radius in Terms of (h, k)
The circle is tangent to the lines x = 3 and y = 2 . For a circle with center (h, k) , the perpendicular distance from the center to each tangent line must equal the radius r .
The distance from the center (h, k) to the line x = 3 is |3 - h| .
The distance from the center (h, k) to the line y = 2 is |2 - k| .
Because the circle is tangent to both lines, these distances (from the center to each line) must be equal and must represent the circle’s radius r .
Therefore,
r = 3 - h = 2 - k.
We only take 3 - h and 2 - k (without absolute signs) because in the first quadrant and under the given conditions, h \lt 3 and k \lt 2 .
Step 4: Relate h and k Using h + k = 2
Since k = 2 - h , substitute this into r = 2 - k :
r = 2 - (2 - h) = h.
Hence, from r = 3 - h and r = h , we get
3 - h = h \quad \Rightarrow \quad 2h = 3 \quad \Rightarrow \quad h = \frac{3}{2}.
Then,
k = 2 - h = 2 - \frac{3}{2} = \frac{1}{2}.
Step 5: Find the Radius and Then the Diameter
The radius r is
r = 3 - h = 3 - \frac{3}{2} = \frac{3}{2}.
Therefore, the diameter D is twice the radius:
D = 2r = 2 \times \frac{3}{2} = 3.
Step 6: Final Answer
The diameter of the required circle is 3.
