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The equation of a circle is x2 + y2 − 4x + 6y + 4 = 0. On the grid provided below, draw the circle.

Mathematics

ScanSolve AI Answer

Answer

(2, -3)
3

Step-by-step solution

5 steps

To draw the circle, we first need to find its center and radius from the given equation. We will convert the general form of the equation to the standard form (x−h)2+(y−k)2=r2(x-h)^2 + (y-k)^2 = r^2 by completing the square.

The given equation is:

x2+y2−4x+6y+4=0x^2 + y^2 - 4x + 6y + 4 = 0
  1. Group the xx terms and yy terms, and move the constant to the right side of the equation.

    (x2−4x)+(y2+6y)=−4(x^2 - 4x) + (y^2 + 6y) = -4
  2. Complete the square for the xx terms.

    Take half of the coefficient of xx (which is −4-4), square it ((−2)2=4(-2)^2 = 4), and add it to both sides.

    (x2−4x+4)+(y2+6y)=−4+4(x^2 - 4x + 4) + (y^2 + 6y) = -4 + 4
  3. Complete the square for the yy terms.

    Take half of the coefficient of yy (which is 6), square it ((3)2=9(3)^2 = 9), and add it to both sides.

    (x2−4x+4)+(y2+6y+9)=−4+4+9(x^2 - 4x + 4) + (y^2 + 6y + 9) = -4 + 4 + 9
  4. Rewrite the expressions in parentheses as squared terms and simplify the right side.

    (x−2)2+(y+3)2=9(x - 2)^2 + (y + 3)^2 = 9
  5. Identify the center (h,k)(h,k) and radius rr.

    Comparing this to the standard form (x−h)2+(y−k)2=r2(x-h)^2 + (y-k)^2 = r^2:
    The center (h,k)(h,k) is (2,−3)(2, -3).
    The radius squared r2r^2 is 9, so the radius r=9=3r = \sqrt{9} = 3.

    To draw the circle on a grid, you would:

    1. Locate the center point at (2,−3)(2, -3).
    2. From the center, measure out 3 units in all directions (up, down, left, right) to mark points on the circle.
    3. Draw a smooth curve connecting these points to form the circle.

    The center of the circle is (2,−3)\boxed{(2, -3)} and the radius is 3\boxed{3}.

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