Question

The locus of centers of all circles passing through two given points A and B is ______.

• A. the bisector of the line segment AB.
• B. the perpendicular bisector of the line segment AB.
• C. any line perpendicular to the line segment AB.
• D. a line parallel to the line segment AB.

Answer 1

The correct option is B

the perpendicular bisector of the line segment AB.

Let P and Q be the centers of two circles S and S’, each passing through two given points A and B. Then,

PA = PB [Radii of the same circle]

$\Rightarrow$ P lies on the perpendicular bisector of AB …(i)

Again, QA = QB [Radii of the same circle]

$\Rightarrow$ Q line on the perpendicular bisector of AB …(ii)

From (i) and (ii), it follows that P and Q both lie on the perpendicular bisector of AB.

Therefore, the locus of the centers of all the circles passing through A and B is the perpendicular bisector of AB.

Hence, the statement is correct.