Let `a ,b ,c` be distinct non-negative numbers and the vectors `a hat i+a hat j+c hat k , hat i+ hat k ,c hat i+c hat j+b hat k` lie in a plane, and then prove that the quadratic equation `a x^2+2c x+b=0` has equal roots.

Explanation

Vectors `ahati+ahatj+chatk, hati+hatk, chati+chatj+bhatk` are coplanar. Thus,
`" "|{:(a,,a,,c), (1,,0,,1),(c,,c,,b):}|=0`
or `" "c^(2)-ab=0`
For equation `ax^(2)+ 2cx+b=0`
`" "D= 4c^(2)- 4ab = 4(c^(2)-ab)=0`
Hence, roots are real and equal.