Question

Given the equation x²+6x+y²+10y=−4, identify the standard form of the equation, the center coordinates, and the radius of the conic section.

Answer 1

The standard form of the given equation is (x+3)² + (y+5)² = 30. The center of the circle is at (-3, -5), and the radius is √30.

Step-by-step explanation:

To find the standard form of a circle from the given equation, we need to complete the square for both the x and y terms. The given equation is x²+6x+y²+10y=−4. Group the x terms together and the y terms together: (x²+6x)+(y²+10y)=−4. Complete the square for each group by adding and subtracting the square of half the coefficient of x and y, respectively.

For the x terms: (x+3)² = x² + 6x + 9, so add and subtract 9.

For the y terms: (y+5)² = y² + 10y + 25, so add and subtract 25.

Add these to both sides of the equation:

(x²+6x+9)+(y²+10y+25)=−4+9+25

This simplifies to:

(x+3)² + (y+5)² = 30

Which is the standard form of a circle. The center coordinates of the circle are (-3, -5), and the radius is the square root of 30.