The correct option is
D None of these
Suppose any line through the given point
(1,−2,3) meets the sphere
x2+y2+z2=4 in the point
(x1,y1,z1)Then x12+y12+z12=4 ...(1)
Now, let the coordinates of the point which divides the join of (1,−2,3) and (x1,y1,z1) in the ration 2:3 be (x2,y2,z2).
Then, we have
x2=2.x1+3.12+3⇒x1=5x2−32
y2=2.y1+3(−2)2+3⇒y1=5y2+62
z2=2.z1+3.32+3⇒z1=5z2−92
Putting values of x1,y1,z1 in (1), we have
(5x2−3)2+(5y2+6)2+(5z2−9)2=4×4
⇒25(x22+y22+z22)−30x2+60y2−90z2+110=0
∴ the locus of (x2,y2,z2) is
5(x2+y2+z2)−6(x−2y+3z)+22=0