A circle in the xy-plane has its center on the line x=3. If the point (4,5) lies on the circle and the radius is sqrt2, which of the following could be the center of the circle? Choose 1 answer: (A) (3,3) (B) (3,4) (c) (3,5) (D) (3,7)

Question

A circle in the  xy-plane has its center on the line  x=3. If the point  (4,5) lies on the circle and the radius is  sqrt2, which of the following could be the center of the circle? Choose 1 answer: (A)  (3,3) (B)  (3,4) (c)  (3,5) (D)  (3,7)

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Circle Distance Formula: Determine the distance formula for a circle. The distance between the center of a circle (h,k) and any point (x,y) on the circle is equal to the radius when using the distance formula: d = sqrt((x-h)^2 + (y-k)^2).Apply Formula to Given Point: Apply the distance formula using the given point (4,5) and the potential centers on the line x=3. Since the x-coordinate of the center must be 3, we only need to find the correct y-coordinate. The radius is given as sqrt2.Check Option (A): Substitute the known values into the distance formula for each option and check if the distance equals sqrt2. Start with option (A) (3,3): d = sqrt((4-3)^2 + (5-3)^2) = sqrt(1^2 + 2^2) = sqrt(1 + 4) = sqrt(5), which is not equal to sqrt2.Check Option (B): Check option (B) (3,4): d = sqrt((4-3)^2 + (5-4)^2) = sqrt(1^2 + 1^2) = sqrt(1 + 1) = sqrt(2), which is equal to sqrt2. This could be the center of the circle.Check Remaining Options: For completeness, check the remaining options. Option (C) (3,5): d = sqrt((4-3)^2 + (5-5)^2) = sqrt(1^2 + 0^2) = sqrt(1), which is not equal to sqrt2. Option (D) (3,7): d = sqrt((4-3)^2 + (5-7)^2) = sqrt(1^2 + (-2)^2) = sqrt(1 + 4) = sqrt(5), which is not equal to sqrt2.

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