From the point (1,−2,3), lines are drawn to meet the sphere x2+y2+z2=4 and they are divided internally in the ratio 2:3. The locus of the point of division is
Suppose, any line l through the given point (1,−2,3) meets the sphere x2+y2+z2=4 in the point (x,y,z).
Then, x21+y21+z21=4 ...(1)
Now, let the coordinates of the point which divides the join of (1,−2,3) and (x1,y1,z1) in the radio 2:3 be (x2,y2,z2).
Then, we have
x2=2.x13.12+3 or x1=5x−32
y2=2.y1+3(−2)2+3 or y1=5y2+62
z2=2.z1+3.32+2 or z1=5y2−92 ....(2)
Putting the values of x1,y1,z1 from (2) 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