The radius of the circle \( x^{2}+y^{2}-2 x+3 y+k=0 \) is \( 2 \frac{1}{2} \). Find the value of...

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The radius of the circle \( x^{2}+y^{2}-2 x+3 y+k=0 \) is \( 2 \frac{1}{2} \). Find the value of \( k \). Find also the equation of the diameter of the circle which passes through the point \( \left(5,2 \frac{1}{2}\right) \).
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