Find the axis, vertex, focus, directrix, equation or the latus rectum, length of the latus rectum of the parabola $x^{2} - 2 x + 8 y + 17 = 0$ and draw the diagram.

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Solution

$x^{2} - 2 x + 8 y + 17 = 0$
$x^{2} - 2 x = - 8 y - 17$
$(x - 1)^{2} - 1 = - 8 y - 17$
$(x - 1)^{2} = - 8 y - 16$
$(x - 1)^{2} = - 8 (y + 2)$
$X^{2} = - 8 Y$ where $X = x - 1, Y = y + 2$
$X^{2} = - 4 (2) Y$
$a = 2$
The type is open downward
Referred to $X, Y$ Referred to $x, y$
$X = x - 1$
$Y = y + 2$
Axis $X = 0$
$X = 0$
$\Rightarrow x - 1 = 0$
$\Rightarrow x - 1$
Vertex $(0, 0)$
$X = 0, Y = 0$
$\Rightarrow x - 1 = 0; y + 2 = 0$
$∴ V (1, - 2)$
Focus $(0, a)$ i.e., $(0, - 2)$
$X = 0; Y = - 2$
$\Rightarrow x - 1 = 0$
$y + 2 = - 2$
$F (1, - 4)$
Directrix
$Y = a$
i.e., $Y = 2$
$Y = 2$ $\Rightarrow y + 2 = 2$
$\Rightarrow y = 0$
Latus
rectum $Y = - a$ i.e.,
$Y = - 2$
$Y = - 2$
$\Rightarrow y + 2 = - 2$
$y = - 4$

Length
$4 a = 8$
$8$

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	Referred to $X, Y$	Referred to $x, y$ $X = x - 1$ $Y = y + 2$
Axis	$X = 0$	$X = 0$ $\Rightarrow x - 1 = 0$ $\Rightarrow x - 1$
Vertex	$(0, 0)$	$X = 0, Y = 0$ $\Rightarrow x - 1 = 0; y + 2 = 0$ $∴ V (1, - 2)$
Focus	$(0, a)$ i.e., $(0, - 2)$	$X = 0; Y = - 2$ $\Rightarrow x - 1 = 0$ $y + 2 = - 2$ $F (1, - 4)$
Directrix	$Y = a$ i.e., $Y = 2$	$Y = 2$ $\Rightarrow y + 2 = 2$ $\Rightarrow y = 0$
Latus rectum	$Y = - a$ i.e., $Y = - 2$	$Y = - 2$ $\Rightarrow y + 2 = - 2$ $y = - 4$
Length	$4 a = 8$	$8$

Find the axis, vertex, focus, directrix, equation or the latus rectum, length of the latus rectum of the parabola x2−2x+8y+17=0 and draw the diagram.

Find the axis, vertex, focus, directrix, equation or the latus rectum, length of the latus rectum of the parabola $x^{2} - 2 x + 8 y + 17 = 0$ and draw the diagram.