Question

Consider a circle $C_1$ with the equation $(x-1{)}^2+y^2=1$ and a shrinking circle $C_2$ with radius $r$, centred at the origin. Let P be the point of intersection of two circles above $x-$axis, $Q$ is the point $(0,r),$ and $R$ is the point of intersection of the line $PQ$ with $x-$axis, then $x-$coordinate of the point $R,$ as $C_2$ shrinks (i.e. $r\to 0$)?

• A. $4$
• B. $2$
• C. $0$
• D. $1$

Answer 1

The correct option is A $4$

$R(x,0),P(0,r),Q(\frac{r^2}{2},\sqrt{r^2-\frac{r^4}{4}})$
$P,Q,R$ are the collinear points.
$\Rightarrow m_{PR}=m_{PQ}$
$\frac{r-0}{0-x}=\frac{r\sqrt{1-\frac{r^2}{4}}-r}{\frac{r^2}{2}-0}$
$\Rightarrow x=\frac{\frac{r^2}{2}}{1-\sqrt{1-\frac{r^2}{4}}}$
$x=\underset{r\to 0}{lim}\frac{\frac{r^2}{2}}{1-\sqrt{1-\frac{r^2}{4}}}=4$