Answer:
Step-by-step explanation:
Given:
- Points (- 3,3) and (3,1) on a circle
- r = 5
i.
There are possible two points that can have a distance of 5 units from both of the given points, so possible two centers, hence two possible circles.
ii.
Let the points are A and B and the centers of circles are F and G.
The midpoint of AB, the point C is:
- C = ((-3 + 3)/2, (3 + 1)/2) = (0, 2)
The length of AB:
- AB = [tex]\sqrt{(3 + 3)^2 + (1 - 3)^2} = \sqrt{6^2+2^2} = \sqrt{40} = 2\sqrt{10}[/tex]
The distance AC = BC = 1/2AB = [tex]\sqrt{10}[/tex]
The distance FC or GC is:
- FC = GC = [tex]\sqrt{5^2-10} = \sqrt{15}[/tex]
Possible coordinates of center are (h, k).
We have radius:
- (h + 3)² + (k - 3)² = 25
- (h - 3)² + (k - 1)² = 25
Comparing the two we get:
- (h + 3)² + (k - 3)² = (h - 3)² + (k - 1)²
Simplifying to get:
We consider this in the distance FC or GC:
- h² + (k - 2)² = 15
- h² + (3h + 2 - 2)² = 15
- 10h² = 15
- h² = 1.5
- h = √1.5 or
- h = - √1.5
Then k is:
- k = 3√1.5 + 2 or
- k = -3√1.5 + 2
So coordinates of centers:
- (√1.5, 3√1.5 + 2) for G or
- (√1.5, 3√1.5 + 2) for F (or vice versa)
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b.
Diameters:
- x - y - 4 = 0
- 2x + 3y + 7 = 0
The intercession of the diameters is the center. We solve the system above and get. Not solving here as it is already a long answer:
The point (2, 4) on he circle given.
Find the radius which is the distance between center and the given point:
- r = [tex]\sqrt{(2 - 1)^2+(4+3)^2} = \sqrt{1^2+7^2} = \sqrt{50}[/tex]
The equation of circle:
