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A circle with center upper C left parenthesis 4 comma negative 2 right parenthesis passes through the point upper A left parenthesis 1 comma 3 right parenthesis. Does the point upper B left parenthesis 8 comma negative 2 right parenthesis lie inside the circle? Prove your answer.

(1 point)
Responses

Point B lies inside the circle since upper C upper A equals Start Root 34 End Root and CB = 4.
Point B lies inside the circle since Image with alt text: upper C upper A equals Start Root 34 End Root and CB = 4.

Point B lies inside the circle since upper C upper A equals Start Root 74 End Root and upper C upper B equals Start Root 32 End Root.
Point B lies inside the circle since Image with alt text: upper C upper A equals Start Root 74 End Root and Image with alt text: upper C upper B equals Start Root 32 End Root .

Point B lies outside the circle since upper C upper A equals Start Root 34 End Root and upper C upper B equals Start Root 74 End Root.
Point B lies outside the circle since Image with alt text: upper C upper A equals Start Root 34 End Root and Image with alt text: upper C upper B equals Start Root 74 End Root .

Point B lies outside the circle since upper C upper A equals Start Root 32 End Root and upper C upper B equals Start Root 34 End Root.

1 answer

To determine whether point \( B(8, -2) \) lies inside the circle centered at \( C(4, -2) \) that passes through point \( A(1, 3) \), we need to follow these steps:

  1. Calculate the radius of the circle: The radius of the circle is the distance from the center \( C \) to point \( A \). We can use the distance formula: \[ C A = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] where \( C(4, -2) \) and \( A(1, 3) \).

    Substituting the coordinates: \[ C A = \sqrt{(1 - 4)^2 + (3 - (-2))^2} = \sqrt{(-3)^2 + (5)^2} = \sqrt{9 + 25} = \sqrt{34} \]

    Therefore, the radius of the circle is \( r = \sqrt{34} \).

  2. Calculate the distance from the center \( C \) to point \( B \): Using the distance formula again: \[ C B = \sqrt{(8 - 4)^2 + (-2 - (-2))^2} = \sqrt{(4)^2 + (0)^2} = \sqrt{16} = 4 \]

  3. Determine the position of point \( B \) relative to the circle: We know:

    • The radius of the circle \( r = \sqrt{34} \approx 5.83 \).
    • The distance from \( C \) to \( B \) is \( 4 \).

    Since \( C B = 4 < \sqrt{34} \), point \( B \) lies inside the circle.

Thus, the correct response would be: Point B lies inside the circle since \( C A = \sqrt{34} \) and \( C B = 4 \).