UPSC NDA Math Model Paper 9

Let, C (x, y, z) be the centre of sphere
AC = BC = CC' = CD = r
CO=√x2+(y−5)2+z2
AO=√(x−4)2+(y−3)2+z2
DO=√(x−4)2+y2+(z−3)2
BO=√x2+(y−4)2+(z−3)2
CO=AO
x2+(y−5)2+z2=(x−4)2 +(y−3)2+z2
⇒x2+y2−10y+25=x2−8x+16 +y2−6y+9
⇒8x−4y=0
⇒2x=y.....(1)
Now, DO=BO
(x−4)2+(y)2+(z−3)2=(x)2 +(y−4)2+(z−3)2
⇒x2−8x+16+y2=x2+y2 −8y+16
⇒8x−8y=0
⇒x=y.....(2)
From (1) and (2),
2x=x⇒x=0So,y=0
AO=DO
16+9+z2=16+z2−6z+9
⇒ 6z=0
⇒ z = 0
∴ Radius =CO=√(0+(5)2+0)
= 5 units
Hence, option (1) is correct.