Prove that the paraboloids x^2/a1^2+y^2/b1^2=2z/c1,x^2/a2^2+y^2/b2^2=2z/c2,x^2/a3^2+y^2/b3^2=2z/c3 have a common tangent plane,if |a1^2 a2^2 a3^2 | |b1^2 b2^2 b3^2 |=0? |c1 c2 c3 |

1 Answer
Jan 17, 2018

See below.

Explanation:

Prove that the paraboloids

#x^2/a_1^2+y^2/b_1^2=(2z)/c_1#
#x^2/a_2^2+y^2/b_2^2=(2z)/c_2#
#x^2/a_3^2+y^2/b_3^2=(2z)/c_3#

have a common tangent plane, if

#det((a_1^2, a_2^2, a_3^2), (b_1^2, b_2^2, b_3^2), (c_1, c_2, c_3 ))=0#

The normal vector to the paraboloid #P_i# is given by

#vec n_i = (x/a_i^2,y/b_i^2,-1/c_i)#

If there exists a common tangent plane then

#vec n_1 = lambda_1 vec n_2# and
#vec n_1 = lambda_2 vec n_3# or

#x/a_1^2= lambda_1 x/a_2^2#
#y/b_1^2=lambda_1 y/b_2^2#
#1/c_1 = lambda 1/c_2#

or

#a_2^2= lambda_1 a_1^2#
#b_2^2=lambda_1 b_1^2#
#c_2 = lambda_1 c_1#

and analogously

#a_3^2= lambda_2 a_1^2#
#b_3^2=lambda_2 b_1^2#
#c_3 = lambda_2 c_1#

this means that the columns in the matrix

#((a_1^2, a_2^2, a_3^2), (b_1^2, b_2^2, b_3^2), (c_1, c_2, c_3 ))#

are linearly dependent so

#det((a_1^2, a_2^2, a_3^2), (b_1^2, b_2^2, b_3^2), (c_1, c_2, c_3 ))=0#