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Find the focus and directrix of the parabola y= 1/2 (x+2)^2 - 3 A. The focus is at (–2,–2) and the directrix is at y = –4. B. The focus is at (–2,–3) and the directrix is at y = –5. C.The focus is at (–2,–2 1/2) and the directrix is at y = –3 1/2. D. The focus is at (–2,–1 1/2) and the directrix is at y = –2 1/2.

User Vince VD
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Answer:

C. The focus is at
(-2,-2 (1)/(2)) and the directrix is at
y = -3\frac{1}2.

Explanation:

First of all, let us learn about the formula to find Focus and Directrix from the standard equation of a parabola.

Standard form of a parabola is given as:


(x - h)^2 = 4p (y - k)

where the focus is
(h, k + p) and

the directrix is
y = k - p

Now, we are given the equation of parabola as:


y= \frac{1}2 (x+2)^2 - 3

Let us try to convert it to standard form:


\Rightarrow y+3= \frac{1}2 (x+2)^2 \\\Rightarrow 2(y+3)= (x+2)^2 \\\Rightarrow (x+2)^2 = 2(y+3)\\\Rightarrow (x+2)^2 = 4* (1)/(2)(y+3)

Comparing the above with standard equation of parabola
(x - h)^2 = 4p (y - k):


h = -2, p = (1)/(2), k = -3

So, Focus is at
(h, k + p)


k + p = -3+(1)/(2) = -2(1)/(2)

Focus is at
(-2, -1\frac{1}2).

Equation of directrix:


y = k - p=-3-(1)/(2)\\\Rightarrow y = -3(1)/(2)

Also, please refer to the attached image for the diagram of given parabola, focus and directrix.

So, the answer is:

C. The focus is at
(-2,-2 (1)/(2)) and the directrix is at
y = -3\frac{1}2.

Find the focus and directrix of the parabola y= 1/2 (x+2)^2 - 3 A. The focus is at-example-1
User Giuliolunati
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7.2k points

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