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Write the equation in standard form for the circle y^2=-x^2-14y

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Answer:

(y + 7)^2 = 98

Explanation:

To write the equation in standard form for the given circle, we need to complete the square for both the x and y variables.

Let's start by rearranging the equation:

y^2 = -x^2 - 14y

Next, we'll complete the square for the y variable. To do this, we need to add a constant term that will complete the square on the left side of the equation. We'll add (14/2)^2 = 49 to both sides:

y^2 + 14y + 49 = -x^2 - 14y + 49

Now, let's group the terms:

y^2 + 14y + x^2 = -14y - x^2 + 49

To simplify the equation further, we can combine the x^2 term and the -x^2 term:

x^2 - x^2 + y^2 + 14y = 49

The x^2 terms cancel each other out, and we are left with:

y^2 + 14y = 49

To complete the square for the y variable, we need to add (14/2)^2 = 49 to both sides:

y^2 + 14y + 49 = 49 + 49

Simplifying further:

(y + 7)^2 = 98

Now, we can write the equation in standard form:

(y + 7)^2 = 98

Therefore, the equation in standard form for the circle is (y + 7)^2 = 98.

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