Electric charges question

It has more than one correct answers

A charged particle of unit mass and charge moves with v=8i+6j m/s in a magnetic field B=2k T:
a)path of particle maybe x^2 + y^2 -4x -24 = 0
b)path of particle maybe x^2 + y^2 - 25 =0
c)path of particle maybe y^2 + z^2 -35 = 0
d) time period = 3.14 sec

and please explain also!!!!

|v|² = 8² + 6² so |v| = 10m/s

The velocity has no z-component. B has only a z-component. So v is perpendicular to B. The particle therefore moves in a circle in the -y plane. The centripetal force, mv²/r, equals the magnetic force, Bqv.
mv²/r = Bqv
Since m = 1 and q = 1
v²/r = Bv
r = v/B = 10 / 2 = 5m

So the correct path must be a circle radius 5m in the xy plane.

In a) Rearrange the equation (complete the square) to check the radius:
x² - 4x + y² - 24 = 0
(x-2)² - 4 + y² - 24 = 0
(x-2)² + y² = 28. This is a circle radius √28, a) is incorrect.

In b) x² + y² = 25 This is a circle radius 5, so b) is correct.

In c) y² + z² - 35 = 0 is the equation of a circle in the yz plane, radius √35. c) is incorrect.

In d) T = distance/speed = circumference / v = 2pi x 5 / 10 =3.14s. d) is correct.