Question

In circle O, G is a moving point on diameter AB. AA' is drawn perpendicular to AB and equal to AG. BB' is drawn perpendicular to AB, on the same side of diameter AB as AA', and equal to $\overline{BG}$. Let O' be the midpoint of A'B'. Then, as G moves from A to B, point O':

• A. moves on a straight line parallel to AB
• B. remains stationary
• C. moves on a straight line perpendicular to AB
• D. moves in a small circle intersecting the given circle
• E. follows a path which iss neither a circle nor a straight line

Answer 1

The correct option is B remains stationary

Let the circle be $x^2+y^2=a^2$

Let G be (h, k)

$\therefore$ k=0, -a<h<a

$\therefore$ G = (h, 0)

AG = AA', AA' $\perp$ AB

$\therefore$ A' = (a, a-h)

BG = BB', BB' $\perp$ AB

$\therefore$ B' = (-a, a+h)

$\therefore$ O' =$\frac{A^{\prime }+B^{\prime }}{2}$ = (0, a)

$\therefore$ Point O' is always constant, i.e., remains stationary (as a is constant).