Question

Question 8
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.

Answer 1

False
Given that PQ is any line segment and $S_1,S_2,S_3,S_4$……… circles are touch a line segment PQ at a point A .
Let the centres of the circles $S_1,S_2,S_3,S_4$ ………..be $C_1,C_2,C_3,C_4$ …….respectively.

To Prove: Centres of these circles lie on the perpendicular bisector of PQ.
Now, joining each centre of the circles to the point A on the line segment PQ by a line segment .i.e $C_{1}A,C_{2}A,C_{3}A,C_{4}A......$ so on.
We know that, if we draw a line from the centre of a circle to its tangent line, then the line is always perpendicular to the tangent line, But it does not bisect the line segment PQ.
$\begin{array}{cc}{C}_{1}A\perp PQ& \left[For{S}_1\right]\\ C_{2}A\perp PQ& \left[For{S}_2\right]\\ C_{3}A\perp PQ& \left[For{S}_3\right]\\ C_{4}A\perp PQ& \left[For{S}_4\right]\end{array}$
Since, each circle is passing through a point A . therefore, all the line segments $C_{1}A,C_{2}A,C_{3}A,C_{4}A......$, so on.
So, the centre of each circle lies on the perpendicular line of PQ but they do not lie on the perpendicular bisector of PQ.
Hence, a number of circles touch a given line segment PQ at a point A, then their centres lie on the perpendicular of PQ but not on the perpendicular bisector of PQ.