If x=r sec α cos β, y=r sec α sin β and z=r tan α, prove that
x2+y2−z2=r2 .
Answer:
- Substituting the values of x, y, and z in x2+y2−z2=r2, we have x2+y2−z2=r2 sec2 α cos2 β+r2 sec2 α sin2 β−r2 tan2 α=r2 sec2 α (cos2 β+sin2 β)−r2 tan2 α=r2 sec2 α−r2 tan2 α[∵cos2 β+sin2 β=1]=r2 (sec2 α−tan2 α)=r2 (1)[∵sec2 α−tan2 α=1]=r2
- Hence, x2+y2−z2 = r2 .