Question

If
$x ^2 + {y }^{2} - 2x + 4y + 1 = 0$
complete one revolution the x-axis then the center in new position is​

Answer 1

Topic

• Coordinate Geometry- Graphs of Circles

$x^{2}+y^{2}-2x+4y+1=0$, the graph of this equation is a circle. To find the center, we first change the form.

$\implies (x-1)^{2}+(y+2)^{2}=2^{2}$

The circle is centered at $(1,-2)$ and its radius is 2.

• Geometry- Circles

If a circle finishes a revolution, the center of the circle moved by its circumference. $\bold{[1]}$

Solution

The circle has a radius of $2$, and the center at $(1,-2)$.

The circle has a diameter of $4\pi$, so is the moved distance of the center.

Now, the circle starts moving. $\bold{[1]}$

After the circle finished one rotation, the center moved by $4\pi$.

Since the direction is not given, the new position of the center will be $(1-4\pi ,-2)$ or $(1+4\pi ,-2)$.

More Information

When a circle does a revolution, the locus of any points of the circle is called the cycloid.

$\bold{[1]}$ Attachment included.