Question

a circle passes through the points o(0,0), a(8,4) and b(6,6). show that oa is a diameter of the circle and find the equation of the circle

Answer 1

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Answer:

(x -4)² +(y -2)² = 20

Explanation:

The circle is the circumcircle of triangle OAB. Then OA will be a diameter if angle OBA is a right angle. We can check that by looking at the slopes of AB and BA.

B-O = (6, 6)-(0, 0) = (6, 6) ⇒ slope = Δy/Δx = 6/6 = 1

A-B = (8, 4) -(6, 6) = (2, -2) ⇒ slope = -2/2 = -1

These slopes have a product of -1, so OB ⊥ BA. The hypotenuse (OA) of ΔOBA is the diameter of the circle.

Then the midpoint of the hypotenuse is the center of the circle.

C = (O +A)/2 = ((0, 0) +(8, 4))/2 = (4, 2)

The equation is ...

(x -h)² +(y -k)² = r² . . . . . circle with center (h, k) and radius r

The radius is the distance from O to C, so is ...

r² = 4² +2² = 20

and the equation of the circle is ...

(x -4)² +(y -2)² = 20 . . . . equation of the circle