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A circle in the xy-plane has the equation x^(2)+y^(2)-6x-10 y=2. What is the diameter of the circle?

A circle in the xy x y -plane has the equation x2+y2−6x−10y=2 x^{2}+y^{2}-6 x-10 y=2 . What is the diameter of the circle?

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Q. A circle in the xy x y -plane has the equation x2+y2−6x−10y=2 x^{2}+y^{2}-6 x-10 y=2 . What is the diameter of the circle?
  1. Write Equation: Write down the given equation of the circle.\newlineThe given equation is x2+y2−6x−10y=2x^{2} + y^{2} - 6x - 10y = 2. We need to complete the square for both xx and yy to bring the equation into the standard form of a circle's equation, which is (x−h)2+(y−k)2=r2(x - h)^{2} + (y - k)^{2} = r^{2}, where (h,k)(h, k) is the center of the circle and rr is its radius.
  2. Rearrange and Group: Rearrange the equation and group xx's and yy's together.\newlineRearrange the equation as follows: x2−6x+y2−10y=2x^{2} - 6x + y^{2} - 10y = 2. This step is necessary to prepare for completing the square for both xx and yy terms.
  3. Complete X Square: Complete the square for the x terms.\newlineTo complete the square for x2−6xx^{2} - 6x, we take half of the coefficient of xx, which is −6/2=−3-6/2 = -3, square it, (−3)2=9(-3)^{2} = 9, and add and subtract it inside the equation. So, we get x2−6x+9−9x^{2} - 6x + 9 - 9.
  4. Complete Y Square: Complete the square for the y terms.\newlineSimilarly, for y2−10yy^2 - 10y, we take half of the coefficient of yy, which is −10/2=−5-10/2 = -5, square it, (−5)2=25(-5)^2 = 25, and add and subtract it inside the equation. So, we get y2−10y+25−25y^2 - 10y + 25 - 25.
  5. Rewrite and Simplify: Rewrite the equation with completed squares and simplify.\newlineAfter completing the squares, the equation becomes x2−6x+9+y2−10y+25=2+9+25x^{2} - 6x + 9 + y^{2} - 10y + 25 = 2 + 9 + 25. Simplify the right side: 2+9+25=362 + 9 + 25 = 36. The equation now is (x−3)2+(y−5)2=36(x - 3)^{2} + (y - 5)^{2} = 36.
  6. Identify Radius: Identify the radius of the circle.\newlineFrom the equation (x−3)2+(y−5)2=36(x - 3)^{2} + (y - 5)^{2} = 36, we see that the right side, 3636, represents r2r^{2}, where rr is the radius of the circle. Therefore, r=36=6r = \sqrt{36} = 6.
  7. Calculate Diameter: Calculate the diameter of the circle.\newlineThe diameter of a circle is twice its radius. Therefore, the diameter =2×r=2×6=12= 2 \times r = 2 \times 6 = 12.

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