Question

The point (1, 4) lies on a circle that is centered at (1, 1). Which statements are correct? Check all that apply.

The circle’s radius is 2 units.
The circle’s radius is 3 units.
The circle’s radius is 9 units.
The point (–2, 1) lies on the circle.
The point (3, 3) lies on the circle.

Answer 1

the circle's radius is 3 units

the point (- 2, 1 ) lies on the circle

the equation of a circle in standard form is

(x - a)² + (y - b)² = r²

where (a, b) are the coordinates of the centre and r is the radius

the radius is the distance from the centre to the point (1, 4 ) on the circle

using (1, 4) and (1,1) in the distance formula, then

r = √(1 - 1 )² + (1 - 4)² = √(0 + 9) =√9 = 3 ⇒ r² = 9

(x - 1 )² + (y - 1 )² = 9 ← equation of circle

substitute the given points into the equation and if equation is true then they lie on the circle

(- 2, 1 ) : (- 2 - 1 )² + (1 - 1 )² = 9 + 0 = 9 ← true

Hence (- 2, 1 ) lies on the circle

(3, 3 ) : (3 - 1 )² + (3 - 1 )² = 4 + 4 = 8 ≠ 9

(3, 3 ) does not lie on the circle

Answer 2

The point (1, 4) lies on a circle that is centered at (1, 1). The circle's radius is 3 units.

Equation of a circle

Let's get to the answer using the formula for the equation of the line at the center of a circle.

The equation of a circle is expressed as:

$(x - h) ^2 +. (y - k) ^ 2 = r ^2$

Where:

(h, k) = The center of the circle

(x, y) = Some points around the circle

(r) = Radius of the circle

Let's apply the formula to solve the question:

$(x - h) ^2 +. (y - k) ^ 2 = r ^2$

$(1 - 1)^2 + (4 - 1)^2 = r^2$

$0^2 + 3^2 = r^2$

$0 + 9 = r^2$

$r^2 = 9$

$r = \sqrt{9}$

r = 3

The circle's radius is 3 units.

Therefore, the correct answer is option b) The circle’s radius is 3 units.

To learn more about the radius of the circle

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