The equation of parabola whose vertex $(2, 5)$ and focus $(2, 2)$ is

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${(x - 2)}^{2} = - 12 (y - 2)$

${(x - 2)}^{2} = 12 (y - 2)$

${(x - 2)}^{2} = 12 (y - 5)$

${(x - 2)}^{2} = - 12 (y - 5)$

answer is B.

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Detailed Solution

$a = A S = \sqrt{{(2 - 2)}^{2} + {(2 - 5)}^{2}}$

$a = 3$

X coordinates of A and S are equal

So $\bar{A S}$ as axis of the parabola and which is parallel to y – axis and

‘S’ is the below the vertex of ‘A’

The equation of parabola is of the from

$\begin{array}{l} {(x - α)}^{2} = - 4 a (y - β) \\ {(x - 2)}^{2} = - 4 (3) (y - 5) \\ {(x - 2)}^{2} = - 12 (y - 5) \end{array}$

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