Final answer:
The correct answer is that the point (-5, -3) is not on the circle because the distance from the circle's center is greater than the radius calculated using the distance formula.
Step-by-step explanation:
The student asked whether the point (-5, -3) is on the circle with the center at (-1, 1) given that the point (4, 2) is on the circle. To answer this, we must compare the distances from the center to each of the points. The radius of the circle is the distance from the center to a point located at the circumference of the circle, which can be found using the distance formula √((x_2 - x_1)^2 + (y_2 - y_1)^2).
First, we find the radius using the given point (4, 2) and the center (-1, 1).
Radius = √((4 - (-1))^2 + (2 - 1)^2) = √(25 + 1) = √26.
Next, we calculate the distance between the center and the point (-5, -3).
Distance = √((-5 - (-1))^2 + (-3 - 1)^2) = √(16 + 16) = √32.
Since √32 is greater than √26, the distance from the center to (-5, -3) is greater than the radius, verifying that the point (-5, -3) is NOT on the circle. Therefore, the correct answer is (a) The distance between the center and (-5, -3) is greater than the radius.