What will be the coordinates of focus of translated parabola $y^{2} - 2 y + 17 = 8 x$ ?

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Solution

Given equation of parabola $y^{2} - 2 y + 17 = 8 x$
On re-arranging the above equation we get,
$\Rightarrow y^{2} - 2 y + 17 = 8 x$
$\Rightarrow y^{2} - (2 \times 1 \times y) + 1 + 16 = 8 x$
$\Rightarrow (y - 1)^{2} = 8 x - 16$
$\Rightarrow (y - 1)^{2} = 4 \times 2 (x - 2)$
Now, on comparing with general equation of translated parabola $(y - k)^{2} = 4 a (x - h)$ we get $k = 2, h = 1,$ and $a = 2.$
$∴$ The coordinates of focus are $(a + h, k)$ i.e. $(3, 2) .$

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