Question

which equation represents a circle that contains the point (-2, 8) and has a center at (4, 0)?
distance formula: \\( \sqrt { ( x _ { 2 } - x _ { 1 } ) ^ { 2 } + ( y _ { 2 } - y _ { 1 } ) ^ { 2 } } \\)
\\( \bigcirc ( x - 4 ) ^ { 2 } + y ^ { 2 } = 100 \\)
\\( \bigcirc ( x - 4 ) ^ { 2 } + y ^ { 2 } = 10 \\)
\\( \bigcirc x ^ { 2 } + ( y - 4 ) ^ { 2 } = 10 \\)
\\( \bigcirc x ^ { 2 } + ( y - 4 ) ^ { 2 } = 100 \\)

Explanation:

Step1: Calculate the radius

Use the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\). Here, \((x_1,y_1)=(4,0)\) (center) and \((x_2,y_2)=(-2,8)\) (point on the circle).

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Step2: Write the equation of the circle

The standard form of the equation of a circle is \((x - h)^2+(y - k)^2=r^2\), where \((h,k)\) is the center and \(r\) is the radius. Here, \(h = 4,k = 0,r = 10\).
So the equation is \((x - 4)^2+y^2=10^2=(x - 4)^2+y^2 = 100\)

Answer:

\((x - 4)^2+y^2 = 100\) (First option)