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Write the equation in standard form. Then identify the center and radius. 14. x^(2)+y^(2)-4x-12 y-129=0

Write the equation in standard form. Then identify the center and radius.\newline1414. x2+y2−4x−12y−129=0 x^{2}+y^{2}-4 x-12 y-129=0

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Q. Write the equation in standard form. Then identify the center and radius.\newline1414. x2+y2−4x−12y−129=0 x^{2}+y^{2}-4 x-12 y-129=0
  1. Rewrite Equation: Step 11: Rewrite the given equation in a more recognizable form.\newlineOriginal Equation: x2+y2−4x−12y−129=0x^2 + y^2 - 4x - 12y - 129 = 0\newlineRearrange terms: x2−4x+y2−12y=129x^2 - 4x + y^2 - 12y = 129
  2. Complete the Square: Step 22: Complete the square for the xx and yy terms.\newlineFor xx: (x2−4x)→(x−2)2−4(x^2 - 4x) \rightarrow (x - 2)^2 - 4\newlineFor yy: (y2−12y)→(y−6)2−36(y^2 - 12y) \rightarrow (y - 6)^2 - 36
  3. Substitute and Simplify: Step 33: Substitute back into the equation and simplify.\newline(x−2)2−4+(y−6)2−36=129(x - 2)^2 - 4 + (y - 6)^2 - 36 = 129\newline(x−2)2+(y−6)2−40=129(x - 2)^2 + (y - 6)^2 - 40 = 129\newline(x−2)2+(y−6)2=169(x - 2)^2 + (y - 6)^2 = 169
  4. Identify Center and Radius: Step 44: Identify the center and radius from the standard form.\newlineStandard form: (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2\newlineHere, h=2h = 2, k=6k = 6, r2=169r^2 = 169\newlineRadius r=169=13r = \sqrt{169} = 13

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