Question

find the locus of the centre of a variable sphere which passes through the origin and meets the axes in A B C so that the volume of the tetrahedron OABC is constant

Answer 1

Answer:

The equation for any sphere that passes through the origin is given as,

x² + y² + z² + 2gx + 2fy + 2kz = 0 ….. (i)

where (-g, -f, -k) is the centre of the sphere

step 1: the point where it cuts the x-axis at A

here, y = 0 and z= 0  

putting y=0 and z=0 in eq. (i), we get

x² + 2gx = 0

or, x(x+2g) = 0

or, x = 0 or x = -2g

∴ the point of intersection = (-2g, 0, 0)

step 2: the point where it cuts the y-axis at B

here, x = 0 and z= 0  

putting x=0 and z=0 in eq. (i), we get

y² + 2fy = 0

or, y(y+2f) = 0

or, y = 0 or y = -2f

∴ the point of intersection = (0, -2f, 0)

step 3: the point where it cuts the z-axis at C

here, x = 0 and y= 0  

putting x=0 and y=0 in eq. (i), we get

z² + 2kz = 0

or, z(z+2k) = 0

or, z = 0 or z = -2k

∴ the point of intersection = (0, 0, -2k)

Step 4:

Now, by joining the point of intersections at A, B & C, we get a tetrahedron, whose each angle is equal to 90° as shown in the figure below.

Therefore,  

The volume of the tetrahedron,

V = [1/6] * [ABC]

⇒ V = 1/6 * [-2g * -2f * -2k]

⇒ V = -8 [gfk] / 6

⇒ -g * -f * -k = 6V/8 = 3V/4  

Since it is given that the volume of tetrahedron is contant, so “3V/4” = c, a constant.  

∴ -g * -f * -k = c

∵ (-g, -f, -k) is the centre of the sphere

∴ x * y * z = c

Thus, the locus of the centre of a variable sphere is,  x * y * z = c, when volume is given to be constant .