If a number of circles touch a given line segment PQ at a point A, then their centers lie on the perpendicular bisector of PQ.

Question

Write ‘True’ or ‘False’ and justify your answer:

If a number of circles touch a given line segment PQ at a point A, then their centers lie on the perpendicular bisector of PQ.

Answer 1

False

Let the S₁, S₂, S₃, …., S_n be n circles with centers C₁, C₂, C_3, …, C_n respectively.

And The PQ is a common tangent to all the circles at point A which is common to all circles.

As we know,

tangent at any point on the circle is perpendicular to the radius through point of contact

we have,

C₁A ⏊ PQ

C₂A ⏊ PQ

C₃A ⏊ PQ

C_nA ⏊ PQ

So, C₁ C₂ C₃ … C_n all lie on the perpendicular line to PQ but not on perpendicular bisector as PA may or may not be equal to AQ .