If a circle has a diameter with end points: (4 + 6i) and (-2 + 6i), 1. Show me how you would determine the length of the diameter and radius.2. Show me how you would determine the center of the circle.3. Determine, mathematically, if (1+9i) lies on the circle. Show how you proved it mathematically.4. Determine, mathematically, if (2-i) lies on the circle. Show how you proved it mathematically.

Answer:Part 1) The diameter is [tex]D=6\ units[/tex]   and the radius is equal to [tex]r=3\ units[/tex]Part 2) The center of the circle is (1+6i)Part 3) The point (1+9i) lies on the circlePart 4) The point (2-i) does not lies on the circleStep-by-step explanation:Part 1) Show me how you would determine the length of the diameter and radius.we have thatThe circle has a diameter with end points: (4 + 6i) and (-2 + 6i)we know thatThe distance between the end points is equal to the diameterthe formula to calculate the distance between two points is equal to[tex]d=\sqrt{(y2-y1)^{2}+(x2-x1)^{2}}[/tex](4 + 6i) ----> (4,6)(-2 + 6i) ---> (-2,6)substitute the values[tex]d=\sqrt{(6-6)^{2}+(-2-4)^{2}}[/tex]    [tex]d=\sqrt{(0)^{2}+(-6)^{2}}[/tex]    [tex]d=6\ units[/tex]    thereforeThe diameter is [tex]D=6\ units[/tex]    The radius is equal to [tex]r=6/2=3\ units[/tex]    ---> the radius is half the diameter Part 2) Show me how you would determine the center of the circlewe know thatThe center of the circle is equal to the midpoint between the endpoints of the diameterThe circle has a diameter with end points: (4 + 6i) and (-2 + 6i)The formula to calculate the midpoint between two points is equal to[tex]M(\frac{x1+x2}{2},\frac{y1+y2}{2})[/tex]substitute[tex]M(\frac{4-2}{2},\frac{6+6}{2})[/tex][tex]M(1,6})[/tex]therefore(1,6) ----> (1+6i)The center of the circle is (1+6i)Part 3) Determine, mathematically, if (1+9i) lies on the circle. Show how you proved it mathematically Find the equation of the circle[tex](x-h)^{2}+(y-k)^{2}=r^{2}[/tex]we haveThe center is (1+6i) -----> (1,6)r=3 unitssubstitute[tex](x-1)^{2}+(y-6)^{2}=3^{2}[/tex][tex](x-1)^{2}+(y-6)^{2}=9[/tex]Verify if the point (1+9i) lies on the circleRemember thatIf a point lies on the circle, then the point must satisfy the equation of the circleSubstitute the value of x and the value of y in the equation and then compare the resultswe havethe point (1+9i) -----> (1,9)[tex](1-1)^{2}+(9-6)^{2}=9[/tex][tex](0)^{2}+(3)^{2}=9[/tex][tex]9=9[/tex] -----> is truethereforeThe point (1+9i) lies on the circlePart 4) Determine, mathematically, if (2-i) lies on the circle. Show how you proved it mathematically The equation of the circle is equal to[tex](x-1)^{2}+(y-6)^{2}=9[/tex]Verify if the point (2-i) lies on the circleRemember thatIf a point lies on the circle, then the point must satisfy the equation of the circleSubstitute the value of x and the value of y in the equation and then compare the resultswe havethe point (2-i) -----> (2,-1)[tex](2-1)^{2}+(-1-6)^{2}=9[/tex][tex](1)^{2}+(-7)^{2}=9[/tex][tex]50=9[/tex] -----> is not truethereforeThe point (2-i) does not lies on the circle