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?
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
#|a_1^2 a_2^2 a_3^2|#
#|b_1^2 b_2^2 b_3^2|#
#|c_1^2 c_2^2 c_3^2|#
#=0#
Here #a_1, b_1, c_1 in RR \ {0}#
- above is a 3x3 column matrix
Prove that the paraboloids
Have a common tangent plane if
Here
- above is a 3x3 column matrix
1 Answer
See below.
Explanation:
If the three paraboloids have a common tangent plane, this tangency can be attained only at
Considering now
or if the tangency is attained over the
resulting in
or
Analogously we get at
Considering now the determinant
if tangency occurs, for instance over the
The same fact occurs if the tangency occurs over the