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A circle graphed in the xy-plane has its center at (0,15). If the point (3,2) lies on the circle, which of the following is an equation of the circle? Choose 1 answer: (A) (x-15)^(2)+y^(2)=178 (B) (x-15)^(2)+y^(2)=sqrt178 (c) x^(2)+(y-15)^(2)=178 (D) x^(2)+(y-15)^(2)=sqrt178

A circle graphed in the xy x y -plane has its center at (0,15) (0,15) . If the point (3,2) (3,2) lies on the circle, which of the following is an equation of the circle?\newlineChoose 11 answer:\newline(A) (x−15)2+y2=178 (x-15)^{2}+y^{2}=178 \newline(B) (x−15)2+y2=178 (x-15)^{2}+y^{2}=\sqrt{178} \newline(C) x2+(y−15)2=178 x^{2}+(y-15)^{2}=178 \newline(D) x2+(y−15)2=178 x^{2}+(y-15)^{2}=\sqrt{178}

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Full solution

Q. A circle graphed in the xy x y -plane has its center at (0,15) (0,15) . If the point (3,2) (3,2) lies on the circle, which of the following is an equation of the circle?\newlineChoose 11 answer:\newline(A) (x−15)2+y2=178 (x-15)^{2}+y^{2}=178 \newline(B) (x−15)2+y2=178 (x-15)^{2}+y^{2}=\sqrt{178} \newline(C) x2+(y−15)2=178 x^{2}+(y-15)^{2}=178 \newline(D) x2+(y−15)2=178 x^{2}+(y-15)^{2}=\sqrt{178}
  1. Identify general form: Identify the general form of the equation of a circle.\newlineThe general form of the equation of a circle with center at (h,k)(h,k) and radius rr is:\newline(x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2.
  2. Fill in center values: Use the center of the circle to fill in the values of hh and kk in the equation.\newlineThe center of the circle is given as (0,15)(0,15), so h=0h = 0 and k=15k = 15. The equation becomes:\newline(x−0)2+(y−15)2=r2(x - 0)^2 + (y - 15)^2 = r^2.\newlineSimplified, this is:\newlinex2+(y−15)2=r2x^2 + (y - 15)^2 = r^2.
  3. Find radius using point: Use the point on the circle to find the radius.\newlineThe point (3,2)(3,2) lies on the circle, so we can substitute x=3x = 3 and y=2y = 2 into the equation to find r2r^2:\newline(3−0)2+(2−15)2=r2(3 - 0)^2 + (2 - 15)^2 = r^2.\newlineCalculate the radius squared:\newline32+(−13)2=r23^2 + (-13)^2 = r^2.\newline9+169=r29 + 169 = r^2.\newline178=r2178 = r^2.
  4. Write final equation: Write the final equation of the circle using the value of r2r^2.\newlineThe final equation of the circle is:\newlinex2+(y−15)2=178x^2 + (y - 15)^2 = 178.

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