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ANSWER

The correct answer is D.

EXPLANATION

The given equation of the circle is
[tex] {x}^{2} + {y}^{2} - 6x + 4y - 51 = 0[/tex]

We rewrite to obtain,

[tex] {x}^{2} - 6x + {y}^{2} + 4y = 51[/tex]

We complete the the square to get,

[tex] {x}^{2} - 6x + ( - 3)^{2}+ {y}^{2} + 4y + {(2)}^{2} = 51 +( - 3)^{2} + {(2)}^{2}.[/tex]

[tex] (x - 3)^{2} + {(y + 2)}^{2} = 51 +9+ 4[/tex]

[tex] (x - 3)^{2} + {(y + 2)}^{2} =64[/tex]

[tex] (x - 3)^{2} + {(y + 2)}^{2} = {8}^{2} [/tex]

Comparing to the equation of the circle in standard form

[tex] (x - h)^{2} + {(y - k)}^{2} = {r}^{2} [/tex]

The centre of the circle is

[tex](h,k)=(3,-2)[/tex]

and radius
[tex]r = 8[/tex]

The correct answer is D.

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