Multiple choice

Circles with radical centre as centre and radius equals to length of tangent from radi- cal centre to any of the three circles will

  1. Bisects the circumference of all the three circles

  2. Bisects the circumference of at least one of

    the circle

  3. Orthogonal to all the three circles

  4. Orthogonal to at least one of the circle
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C Correct answer
Explanation

The radical center of three circles is the point from which the lengths of tangents to all three circles are equal. A circle centered at the radical center with this tangent length as the radius is the radical circle, which is orthogonal to all three circles.

AI explanation

The radical center is the unique point having equal tangent lengths to all three given circles. A circle is orthogonal to another circle if the square of the distance between their centers equals the sum of the squares of their radii. Using the Pythagorean theorem with the distance from the radical center to a given circle's center and the equal tangent length as the orthogonal circle's radius, the new circle is orthogonal to all the three circles.

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