Find the vertex focus, equation of directrix and equation of axis of the $y^{2} - x + 4 y + 5 = 0$ .

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Solution

$\begin{matrix} y^{2} + 4 y - x + 5 = 0 y^{2} + 4 y + 4 = x - 5 {(y + 2)}^{2} = (x - 5) s t a n d a r d f o r m o f e q u a t i o n . {(y - b)}^{2} = 2 P (x - a) ∴ b = - 2 a = 52 P = 1 \Rightarrow p = \frac{1}{2} v e r t e x i s (a, b) \Rightarrow (5, - 2) f o c u s i s (a + \frac{P}{2}, b) \Rightarrow (5 + \frac{1}{\frac{2}{2}}, - 2) = (\frac{21}{4}; - 2) d i r e c t r i x i s x = a - \frac{P}{2} = 5 - \frac{1}{4} = \frac{19}{4} \end{matrix}$

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