Question

the point (1, 4) lies on a circle that is centered at (1, 1). which statements are correct? check all that apply. the circles radius is 2 units. the circles radius is 3 units. the circles radius is 9 units. the point (-2, 1) lies on the circle. the point (3, 3) lies on the circle.

Explanation:

Step1: Calculate the radius

The radius of a circle centered at \((h,k)\) and passing through \((x,y)\) is given by the distance formula \(r = \sqrt{(x - h)^2+(y - k)^2}\). Here, the center is \((1,1)\) and the point on the circle is \((1,4)\). So, \(r=\sqrt{(1 - 1)^2+(4 - 1)^2}=\sqrt{0 + 9}=3\) units. So the radius is 3 units.

Step2: Check if \((-2,1)\) lies on the circle

The distance between \((-2,1)\) and the center \((1,1)\) is \(\sqrt{(-2 - 1)^2+(1 - 1)^2}=\sqrt{(-3)^2+0}=\sqrt{9} = 3\) units, which is equal to the radius. So \((-2,1)\) lies on the circle.

Step3: Check if \((3,3)\) lies on the circle

The distance between \((3,3)\) and the center \((1,1)\) is \(\sqrt{(3 - 1)^2+(3 - 1)^2}=\sqrt{4 + 4}=\sqrt{8}\approx2.83\) units, which is not equal to 3. So \((3,3)\) does not lie on the circle.

Answer:

B. The circle's radius is 3 units.
D. The point \((-2,1)\) lies on the circle.