Total number of circles that can pass through 2 pair of points in a plane are ________.

Question

Accepted Answer

{"content":[{"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step 1 (Geometric Condition):"},{"type":"text","text":" Let the two points in the plane be "},{"type":"math_inline","attrs":{"latex":"A"}},{"text":" and ","type":"text"},{"attrs":{"latex":"B"},"type":"math_inline"},{"text":". A circle passing through both ","type":"text"},{"attrs":{"latex":"A"},"type":"math_inline"},{"text":" and ","type":"text"},{"attrs":{"latex":"B"},"type":"math_inline"},{"text":" must have its center equidistant from ","type":"text"},{"attrs":{"latex":"A"},"type":"math_inline"},{"text":" and ","type":"text"},{"attrs":{"latex":"B"},"type":"math_inline"},{"text":".","type":"text"}],"type":"paragraph"},{"content":[{"marks":[{"type":"bold"}],"text":"Step 2 (Locus of Centers):","type":"text"},{"text":" The set of all points equidistant from ","type":"text"},{"attrs":{"latex":"A"},"type":"math_inline"},{"text":" and ","type":"text"},{"attrs":{"latex":"B"},"type":"math_inline"},{"text":" is the perpendicular bisector of the line segment ","type":"text"},{"attrs":{"latex":"AB"},"type":"math_inline"},{"text":". This bisector is a straight line containing infinitely many points.","type":"text"}],"type":"paragraph"},{"content":[{"text":"Step 3 (Circles Construction):","type":"text","marks":[{"type":"bold"}]},{"text":" Any point ","type":"text"},{"type":"math_inline","attrs":{"latex":"O"}},{"text":" on this perpendicular bisector can be chosen as the center of a circle with radius ","type":"text"},{"attrs":{"latex":"OA"},"type":"math_inline"},{"text":" (which equals ","type":"text"},{"attrs":{"latex":"OB"},"type":"math_inline"},{"text":"). Since there are infinitely many points on the perpendicular bisector, there are infinitely many distinct circles that can pass through the two points.","type":"text"}],"type":"paragraph"},{"content":[{"marks":[{"type":"bold"}],"text":"Step 4 (Conclusion):","type":"text"},{"type":"text","text":" The number of circles is "},{"marks":[{"type":"bold"}],"text":"infinity","type":"text"},{"text":".","type":"text"}],"type":"paragraph"}],"type":"doc"}

Guide / Hint

Solution