Question

Determine the equation of the circle with radius 8 and center (−2,−7).

Answer 1

The equation of the circle with center (-2, -7) and radius 8 is (x + 2)² + (y + 7)² = 64.

Step-by-step explanation:

The equation of a circle in the coordinate plane can be expressed using the formula (x - h)² + (y - k)² = r², where (h, k) is the center of the circle and r is the radius. Given a circle with center at (-2, -7) and a radius of 8, we can substitute these values into the formula to determine its equation.

The equation of the circle is (x + 2)² + (y + 7)² = 64. This follows from the general formula by adding 2 to x and adding 7 to y to account for the shifted center and squaring the radius, 8, to get 64, the radius squared.

Answer 2

Answer:
`\begin{gathered} (x\text{ + 2\rparen}^2\text{ + \lparen y + 7\rparen}^2\text{ = 64} \\ This\text{ is the equation in vertex form} \\ \\ There\text{ is also equation in general form:} \\ x^2\text{ + y}^2+\text{ 4x + 14y - 11 = 0} \end{gathered}`

Step-by-step explanation:

Given:

radius = 8

Center of circle is (-2, -7)

To find:

The equation of the circle using the given information

To determine the equation of the circle we will apply the formula:

`\begin{gathered} (x\text{ - h\rparen}^2\text{ + \lparen y - k\rparen}^2\text{ = r}^2 \\ Th\text{ equation of circle in vertex form} \end{gathered}`

where center (-2, -7) = (h, k)

h = -2, k = -7, r = 8

substitute the values into the formula:

`(x\text{ - \lparen-2\rparen\rparen}^2\text{ + \lparen y -\lparen- 7\rparen\rparen}^2\text{ = 8}^2`

`\begin{gathered} (x\text{ + 2\rparen}^2\text{ + \lparen y + 7\rparen}^2\text{ = 64} \\ This\text{ is the equation in vertex form} \\ \\ There\text{ is also equation in standard form:} \\ (x+2)(x+2)\text{ + \lparen y+7\rparen\lparen y+7\rparen = 64} \\ x^2\text{ + 4x + 4 + y}^2+\text{ 14y + 49 = 64} \\ x^2\text{ + y}^2\text{ + 4x + 14y + 53 - 64 = 0} \\ x^2\text{ + y}^2+\text{ 4x + 14y - 11 = 0} \end{gathered}`

The question did not specify which of the form. That is why the equation is written in vertex or standard form

`\begin{gathered} (x\text{ + 2\rparen}^2\text{ + \lparen y + 7\rparen}^2\text{ = 64} \\ This\text{ is the equation in vertex form} \\ \\ There\text{ is also equation in general form:} \\ x^2\text{ + y}^2+\text{ 4x + 14y - 11 = 0} \end{gathered}`