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.