If a number of circles touch a given line segment \(PQ\) at a point \(A\), then their centres lie on the perpendicular bisector of \(PQ\). State True/False and justify.
False.
Step 1: Recall the property of a tangent. A circle touches a line at only one point. The radius drawn to the point of contact is always perpendicular to the tangent at that point.
Step 2: Here, the line segment is \(PQ\). The circle touches it at a point \(A\). So the centre of the circle must lie on the line passing through \(A\) and perpendicular to \(PQ\).
Step 3: The perpendicular bisector of \(PQ\) is a special line. It is perpendicular to \(PQ\) and passes through the midpoint of \(PQ\).
Step 4: For the circle’s centre to lie on the perpendicular bisector of \(PQ\), the point of contact \(A\) must be exactly the midpoint of \(PQ\). But in general, \(A\) can be anywhere on \(PQ\), not necessarily at the midpoint.
Step 5: Therefore, the centre of such circles lies on the line through \(A\) perpendicular to \(PQ\), not on the perpendicular bisector of \(PQ\) (except in the special case when \(A\) is the midpoint).