A circle is centered at \( I(0,0) \). The point \( T(5,8) \) is on the circle. Where does the point \( Z(-\sqrt{15}, \sqrt{74}) \) lie? Choose 1 answer: (A) Inside the circle (B) On the circle (C) Outside the circle

To find out where point \( Z(-\sqrt{15}, \sqrt{74}) \) lies in relation to the circle centered at \( I(0,0) \) with point \( T(5,8) \) on its circumference, we first determine the radius of the circle. The radius can be calculated using the distance formula: \[ r = \sqrt{(5-0)^2 + (8-0)^2} = \sqrt{25 + 64} = \sqrt{89}. \] Next, we calculate the distance from the center of the circle \( I(0,0) \) to point \( Z(-\sqrt{15}, \sqrt{74}) \): \[ d_Z = \sqrt{(-\sqrt{15} - 0)^2 + (\sqrt{74} - 0)^2} = \sqrt{15 + 74} = \sqrt{89}. \] Since the distance \( d_Z = \sqrt{89} \) is equal to the radius \( r = \sqrt{89} \), we can conclude that point \( Z \) lies on the circle. Thus, the answer is: (B) On the circle.