How do you find the vertex, focus and directrix of $x^2 - 2x + 44y + 353 = 0$ ?

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Answer 1 · 2017-05-16T21:39:15

Vertex: $(1, -8)$
Focus: $(1, -19)$
Directrix: $y = 3$

Start by moving the constant and $y$ -value to the right side of the equation, then complete the square for $x$ :

$(x^2 - 2x + 1) = -44y - 353 + 1$

$(x-1)^2 = -44y - 352$

$-1/44(x-1)^2 = y + 8$

We can now see that the vertex is $(1, -8)$

The focus is $(h, k + 1/(4a))$ , which makes the focus of this equation

$(1, -8 + 1/(4(-1/44))) = (1, -19)$

The directrix of this equation is

$y = k - 1/(4a) = -8 - 1/(4(-1/44)) = 3$

How do you find the vertex, focus and directrix of x^2 - 2x + 44y + 353 = 0 x^2 - 2x + 44y + 353 = 0 ?