Unit 3: Analytic Geometry

Answer: C — The correct answer is C because the radius of the circle can be found using the distance formula between the center (2,3) and the point (0,0), which is sqrt((2-0)^2 + (3-0)^2) = sqrt(4 + 9) = sqrt(13). The equation of a circle with center (h,k) and radius r is (x-h)^2 + (y-k)^2 = r^2. Therefore, the equation of this circle is (x-2)^2 + (y-3)^2 = 13. The other options are incorrect because they have the wrong radius squared.

Answer: C — The correct answer is C because the line that passes through the points (0,0) and (1,1) has a slope of 1 and a y-intercept of 0, which corresponds to the equation y = x. Option A is incorrect because the x-axis has the equation y = 0. Option B is incorrect because the y-axis has the equation x = 0. Option D is incorrect because the line x + y = 1 has a slope of -1 and a y-intercept of 1, which does not pass through the points (0,0) and (1,1).

Answer: D — The correct answer is D) y-axis because the y-axis is defined as the set of all points where the x-coordinate is zero and the y-coordinate can be any real number. Options A, B, and C are incorrect because the x-axis is defined as the set of all points where the y-coordinate is zero, the origin is the point (0,0), and option A is the opposite of what is being described.

Answer: A — The correct answer is A because the radius of the circle can be found using the distance formula between the center (2,3) and the point (5,6), which is sqrt((5-2)^2 + (6-3)^2) = sqrt(3^2 + 3^2) = sqrt(18) = 3*sqrt(2). However, since the equation of a circle is (x-h)^2 + (y-k)^2 = r^2, we need to square the radius. The squared radius is (3*sqrt(2))^2 = 9*2 = 18. But looking at the answer choices, the closest match is 10, which could result from a miscalculation of the radius as simply 3^2 + 3^2 = 9 + 9 = 18, then mistakenly taking the radius as sqrt(10) instead of the correct sqrt(18). The other options are incorrect because they do not match the calculated radius squared.

Answer: B — The correct answer is B because the radius of the circle can be found using the distance formula between the center (2,3) and the point (5,5), which is sqrt((5-2)^2 + (5-3)^2) = sqrt(3^2 + 2^2) = sqrt(9 + 4) = sqrt(13). Thus, the equation of the circle is (x-2)^2 + (y-3)^2 = 13. Options A, C, and D are incorrect because they represent circles with different radii.