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A circle in the xy-plane has its center at (-4,5) and the point (-8,8) lies on the circle. Which equation represents this circle?

A circle in the xy x y -plane has its center at (−4,5) (-4,5) and the point (−8,8) (-8,8) lies on the circle. Which equation represents this circle?

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Full solution

Q. A circle in the xy x y -plane has its center at (−4,5) (-4,5) and the point (−8,8) (-8,8) lies on the circle. Which equation represents this circle?
  1. Identify Center and Point: Identify the center (h, k) and a point on the circle to find the radius.\newlineCenter: (−4-4, 55)\newlinePoint on circle: (−8-8, 88)\newlineCalculate the radius using the distance formula: r=(x2−x1)2+(y2−y1)2 r = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \newliner=((−8)−(−4))2+(8−5)2 r = \sqrt{((-8) - (-4))^2 + (8 - 5)^2} \newliner=(−4)2+32 r = \sqrt{(-4)^2 + 3^2} \newliner=16+9 r = \sqrt{16 + 9} \newliner=25 r = \sqrt{25} \newliner=5 r = 5
  2. Calculate Radius: Substitute the values of h, k, and r into the standard form equation of a circle.\newlineStandard form: (x−h)2+(y−k)2=r2 (x - h)^2 + (y - k)^2 = r^2 \newlineSubstitute h = −4-4, k = 55, and r = 55.\newline(x−(−4))2+(y−5)2=52 (x - (-4))^2 + (y - 5)^2 = 5^2 \newline(x+4)2+(y−5)2=25 (x + 4)^2 + (y - 5)^2 = 25

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