Question

In the figure, $P$ and $Q$ are the points of intersection of two circles with centres $O$ and $O^{\prime }.$ If straight lines, $APB$ and $CQD$ are parallel to $O{O}^{\prime },$ then what is the ratio of $O{O}^{\prime }$ and $AB?$

• A. $1:3$
• B. $1:2$
• C. $1:4$
• D. $3:4$

Answer 1

The correct option is B

$1:2$

Draw $OM\perp AP$ and $O^{\prime }N\perp PB$.

Now, quadrilateral $O{O}^{\prime }NM$ is a rectangle, since, all of its angles are ${90}^{\circ }$

$\Rightarrow O{O}^{\prime }=MN$

We know that, perpendicular from the centre of the circle to the chord bisects the chord, therefore

$AM=MP=\frac{1}{2}AP$ and

$PN=NB=\frac{1}{2}BP$

$\Rightarrow O{O}^{\prime }=MN=PM+PN$
$=\frac{AP}{2}+\frac{BP}{2}$
$=\frac{(AP+BP)}{2}$
$=\frac{AB}{2}$

$\therefore O{O}^{\prime }:AB=1:2$