Identify the conic section whose equation is given:
x2 + 10x + y2 + 12y = 60
Clearly looks like a circle.
I can tell by its form of
x^2 + y^2 + Bx + Cy = D
To identify the conic section represented by the equation x² + 10x + y² + 12y = 60, we can manipulate the equation to transform it into standard form. The standard form of the equation for a conic section depends on the coefficients of x², y², and the constant term.
First, let's complete the square to convert the equation into standard form. To do this, we'll need to add and subtract specific terms on both sides of the equation:
x² + 10x + y² + 12y = 60
Rearranging the equation, we have:
(x² + 10x) + (y² + 12y) = 60
To complete the square for the x-terms, we take half of the coefficient of x (which is 10) and square it:
(10/2)² = 25
We add 25 to both sides of the equation:
(x² + 10x + 25) + (y² + 12y) = 60 + 25
Simplifying further:
(x + 5)² + (y² + 12y) = 85
To complete the square for the y-terms, we take half of the coefficient of y (which is 12) and square it:
(12/2)² = 36
We add 36 to both sides of the equation:
(x + 5)² + (y² + 12y + 36) = 85 + 36
Simplifying further:
(x + 5)² + (y + 6)² = 121
Now that the equation is in standard form as (x + h)² + (y + k)² = r², we can determine the conic section.
Since the x-term and y-term have the same sign and the coefficients of x² and y² are both positive, we have a circle. The center of the circle is (-5, -6), and the radius squared is 121. Hence, the conic section represented by the equation is a circle.