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)

Bytelearn · Accepted Answer

Understand the problem: Understand the problem. We are given a circle with a radius of sqrt2 and a point on the circle (4,5). The center of the circle lies on the line x=3. We need to find which of the given options could be the center of the circle.Use distance formula: Use the distance formula to find the distance between the center (3,y) and the point (4,5). The distance formula is d = sqrt((x2 - x1)^2 + (y2 - y1)^2), where (x1, y1) and (x2, y2) are points on the plane. Here, (x1, y1) is the center of the circle (3, y) and (x2, y2) is the point on the circle (4,5).Plug in values: Plug in the values into the distance formula. We know the radius is sqrt2, so the distance d is sqrt2. Plugging in the values, we get sqrt2 = sqrt((4 - 3)^2 + (5 - y)^2).Simplify equation: Simplify the equation. Squaring both sides to eliminate the square root gives us 2 = (1)^2 + (5 - y)^2.Further simplify: Further simplify the equation. 2 = 1 + (5 - y)^2 leads to 1 = (5 - y)^2.Solve for y: Solve for y. Taking the square root of both sides gives us 1 = 5 - y or y = 5 - 1, which simplifies to y = 4.Check the answer: Check the answer. The center of the circle must be on the line x=3, and we found y=4. So the center of the circle is (3,4), which is option (B).

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