The points B and C with coordinates (5,5) and (-3,-1) respectively are the ends of the diameter of a circle centre A.
a) on the graph draw the circle.
b)determine the equation of the circle expressing it in the form x^2+y^2+ax+by+c where a,b,c are integral values.
a) To draw the circle, we will first find the coordinates of the center of the circle, A. The center of a circle is the midpoint of its diameter, so we can find the coordinates of A by finding the average of the x-coordinates and the average of the y-coordinates of B and C.
x-coordinate of A = (5 + (-3))/2 = 2/2 = 1
y-coordinate of A = (5 + (-1))/2 = 4/2 = 2
So the center of the circle is A(1,2).
The radius of the circle is half the distance between B and C. We can find this distance using the distance formula: √((x2-x1)^2 + (y2-y1)^2).
Distance between B and C = √((5-(-3))^2 + (5-(-1))^2)
= √((5+3)^2 + (5+1)^2)
= √(8^2 + 6^2)
= √(64 + 36)
= √100
= 10
So the radius of the circle is 10.
Now we can plot the circle on the graph. The center is A(1,2) and the radius is 10. Using a compass, draw the circle around point A with a radius of 10. It should intersect the points B and C.
b) To determine the equation of the circle, we can use the general form of the equation of a circle: (x-h)^2 + (y-k)^2 = r^2, where (h,k) is the center of the circle and r is the radius.
In this case, the center of the circle is A(1,2) and the radius is 10.
So the equation of the circle is (x-1)^2 + (y-2)^2 = 10^2.
Expanding and rearranging:
(x^2 - 2x + 1) + (y^2 - 4y + 4) = 100
x^2 + y^2 - 2x - 4y + 5 = 100
x^2 + y^2 - 2x - 4y - 95 = 0
So the equation of the circle is x^2 + y^2 - 2x - 4y - 95 = 0, with a = -2, b = -4, and c = -95.