The length of arc AXB is twice the length of arc BZC, and

The red part is not correct.

When we are told that "the length of arc AXB is twice the length of arc BZC" means that \(\angle{AOB}=2*\angle{BOC}\), where O is the center of the circle. Similarly "the length of arc AYC is three times the length of arc AXB." means that \(\angle{YOC}=3*\angle{AOB}\)

So if \(\angle{BOC}=x\) then \(x+2x+3*2x=360\) --> \(x=40\) --> \(\angle{AOB}=2*\angle{BOC}=2*40=80\). Now, according to The Central Angle Theorem the measure of inscribed angle is always half the measure of the central angle so \(\angle{AOB}=80=2*\angle{BCA}\) --> \(\angle{BCA}=40\).

Answer: B.

To elaborate more on angles and arcs:
You should know that when you measure the length of an arc in degrees then it's the measure of corresponding central angle. For example \(\angle{BOC}=40\) means that arcBZC is 40/360=1/9 th of the circumference. Next, \(\angle{BAC}\), according to the central angle theorem, will be half the measure of the central angle, so \(\angle{BAC}=\frac{\angle{BOC}}{2}=20\). As for \(\angle{BZC}\): it'll be supplementary angle to \(\angle{BAC}\) (supplementary angles are two angles that add up to 180°), so it equals to \(\angle{BZC}=180-\angle{BAC}=160\).

For more chek Circles chapter of Math Book: math-circles-87957.html

Hope it helps.