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Write Equations for Conic Sections

1. Write an equation for the parabola with vertex (1, 1) and directrix x = $\frac{3}{2}$ .

opens horizontally

$x=a(y-k)^{2}+h$ , vertex = (h, k)

$x=a(y-1)^{2}+1$

$directrix, x=h-\frac{1}{4a}$

$\frac{3}{2}=1-\frac{1}{4a}$

$\frac{3}{2}-1=1-\frac{1}{4a}-1$

$\frac{1}{2}=-\frac{1}{4a}$

$4a(\frac{1}{2})=-1$

$\frac{2a}{2}=\frac{-1}{2}$

$a=-\frac{1}{2}$

$x=-\frac{1}{2}(y-1)^{2}+1$

2. Write an equation for the circle centered at (-2. 1) with a radius of 3 units.

$(x-h)^{2}+(y-k)^{2}=r^{2}$ , r = 3 units

$(x-h)^{2}+(y-k)^{2}=3^{2}=9$ , h = -2, k = 1

$(x--2)^{2}+(y-1)^{2}=9$

$(x+2)^{2}+(y-1)^{2}=9$

3. Find me foci for the ellipse $\frac{x^2}{9}+\frac{y^2}{25}=1$

$\frac{x^2}{b^2}+\frac{y^2}{a^2}=1$

foci: (0, c), (0, -c)

$c^{2}=a^{2}-b^{2}$

$a^{2}=25, b^{2}=9$

$c^{2}=25-9$

$c^{2}=16$

$\sqrt{c^2}=\sqrt{16}$

$c=4$

foci: (0, 4), (0, -4)