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A circle in the xyxy-plane has its center at the point (−6,1)(-6,1). If the point (7,12)(7,12) lies on the circle, what is the radius of the circle?

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Full solution

Q. A circle in the xyxy-plane has its center at the point (−6,1)(-6,1). If the point (7,12)(7,12) lies on the circle, what is the radius of the circle?
  1. Identify Center and Point: We have:\newlineCenter: (−6,1)(-6,1)\newlinePoint on the circle: (7,12)(7,12)\newlineIdentify the values of the center (h,k)(h,k) and the point (x,y)(x,y).\newlineh=−6h = -6\newlinek=1k = 1\newlinex=7x = 7\newliney=12y = 12\newlineThe radius rr can be found using the distance formula between the center and the point on the circle.
  2. Distance Formula: The distance formula is given by:\newliner=[(x−h)2+(y−k)2]r = \sqrt{[(x - h)^2 + (y - k)^2]}\newlineSubstitute the values of hh, kk, xx, and yy into the distance formula to find the radius.\newliner=[(7−(−6))2+(12−1)2]r = \sqrt{[(7 - (-6))^2 + (12 - 1)^2]}
  3. Calculate Radius: Calculate the squares and the distance:\newliner=[(7+6)2+(12−1)2]r = \sqrt{[(7 + 6)^2 + (12 - 1)^2]}\newliner=[132+112]r = \sqrt{[13^2 + 11^2]}\newliner=[169+121]r = \sqrt{[169 + 121]}
  4. Add Squares: Add the squares to find the radius:\newliner=169+121r = \sqrt{169 + 121}\newliner=290r = \sqrt{290}
  5. Find Radius: Take the square root to find the radius:\newliner=290r = \sqrt{290}

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