Question

A circle of radius $14\text{units}$ touches the coordinate axes in the first quadrant. If the circle makes three complete rolls along positive direction of $Y-\text{axis}$, then the equation of circle in the new position is

• A. $(x-14{)}^2+(y-278{)}^2=14$
• B. $(x-14{)}^2+(y-278{)}^2=196$
• C. $(x+14{)}^2+(y+278{)}^2=196$
• D. $(x-14{)}^2+(y-102{)}^2=196$

Answer 1

The correct option is B $(x-14{)}^2+(y-278{)}^2=196$

Center of the circle in initial position: $C_1\equiv (14,14)$
When circle is rolled $n$ complete cycles along positive direction of $Y-\text{axis}$,
New position of centre:
$C_2\equiv (14,14+n(2\pi r\left)\right)$
Putting $n=3$, we get
$\Rightarrow C_2\equiv (14,14+6\pi \cdot 14)\Rightarrow C_2\equiv (14,14+3\times 2\times \frac{22}{7}\times 14)\Rightarrow C_2\equiv (14,278)$
The radius of the new circle is same as $r=14\text{units}$

$\therefore$ The equation of the required circle is
$(x-14{)}^2+(y-278{)}^2=196$