Mathematics

Mathematics

Problem

The endpoints of a diameter of a circle are A(5,−1)A(5, -1) and B(−1,3)B(-1, 3). Using this information, answer the following questions. a. (i) Determine the coordinates of the centre of the circle and its radius. b. (ii) Obtain the Cartesian equation of the circle in standard form. c. (iii) Draw a clear sketch of the circle on a coordinate plane and mark the points where it meets the coordinate axes. d. (iv) Verify that the point AA lies on the circle by direct substitution in your equation from part (ii).
Intuitive explanation

The key is recognizing that the center of the circle is the midpoint of any diameter, and the radius is half the diameter's length. Once you find the center and radius, you can write the circle's equation in standard form (x−h)2+(y−k)2=r2(x-h)^2 + (y-k)^2 = r^2. This gives you a complete description of the circle from just two points.

Intuitive explanation

The key is recognizing that the center of the circle is the midpoint of any diameter, and the radius is half the diameter's length. Once you find the center and radius, you can write the circle's equation in standard form (x−h)2+(y−k)2=r2(x-h)^2 + (y-k)^2 = r^2. This gives you a complete description of the circle from just two points.

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