Show that the equation $y^{2} + 6 y - 2 x + 5 = 0$ represents parabola. Find its verses, focus, latus, rectum, the equation of axis and directrix.

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Solution

$y^{2} + 6 y + 5 = 2 x$
$\Rightarrow {(y + 3)}^{2} = 2 (x + 2)$
$y^{2} = 4 A \times$ (Parabola)
On comparing, $A = \frac{1}{2}$
Vertex $(- 2, - 3)$ ; Focii $(\frac{- 3}{2}, - 3)$
Directrix : $x = \frac{- 5}{2}$
Length of Latus Rectum $= 4 A = 2$

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