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, \(r=\sqrt{(1 - 1)^2+(4 - 1)^2}=\sqrt{0 + 9}=\sqrt{9} = 3\) units.

Step2: Check each statement

• For "The circle’s radius is 2 units": Incorrect, as we found radius is 3.
• For "The circle’s radius is 3 units": Correct, from step 1.
• For "The circle’s radius is 9 units": Incorrect, radius is 3.
• For "The point \((- 2,1)\) lies on the circle": Calculate distance from \((-2,1)\) to \((1,1)\): \(d=\sqrt{(-2 - 1)^2+(1 - 1)^2}=\sqrt{9+0}=3\), which is equal to radius, so this point lies on the circle.
• For "The point \((3,3)\) lies on the circle": Calculate distance from \((3,3)\) to \((1,1)\): \(d=\sqrt{(3 - 1)^2+(3 - 1)^2}=\sqrt{4 + 4}=\sqrt{8}

eq3\), so this point does not lie on the circle.

Answer:

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