Question

Which of these points lies on the circle with the center (2,3) and radius 2?

A. (4,3)
B. (-1,0)
C. (1,3)
D. (3,4)

Answer 1

A. (4, 3)

Explanation:

If the point (x, y) lies on a circle, then the distance between this point and the center is equal to the radius.

The formula of a distance between two points:

`d=√((x_2-x_1)^2+(y_2-y_1)^2)`

We have the center in (2, 3) and the radius r = 2.

Check for

`A.\ (4,\ 3)\\\\d=√((4-2)^2+(3-3)^2)=√(2^2+0^2)=√(4+0)=\sqrt4=2\\\\\bold{CORRECT}\\\\B.\ (-1,\ 0)\\\\d=√((-1-2)^2+(0-3)^2)=√((-3)^2+(-3)^2)=√(9+9)=√(18)\\eq2\\\\C.\ (1,\ 3)\\\\d=√((1-2)^2+(3-3)^2)=√((-1)^2+0^2)=√(1+0)=√(1)=1\\eq2\\\\D.\ (3,\ 4)\\\\d=√((3-2)^2+(4-3)^2)=√(1^2+1^2)=√(1+1)=\sqrt2\\eq2`

Answer 2

The answer to your question is the letter A.

Explanation:

Data

Center = (2, 3)

radius = 2

Process

1.- Find the equation of the line

(x - h)² + (y - k)² = r²

-Substitution

(x - 2)² + (y - 3)² = 2²

-Simplification

(x - 2)² + (y - 3)² = 4

-Evaluate the points in the equation

A. (4, 3)

(4 - 2)² + (3 - 3)² = 4

2² + 0 = 4

4 = 4 This point lies in the circle

B (-1, 0)

(-1 - 2)² + (0 - 3)² = 4

-3² + (-3)² = 4

9 + 9 = 4 This point is not part of the circle

C. (1, 3)

(1 - 2)² + (3 - 3)² = 4

(-1)² + (0)² = 4

1 = 4 This point is not part of the circle

D. (3, 4)

(3 - 2)² + (4 - 3)² = 4

1² + 1² = 4

2 = 4 This point is not part of the circle