If `a in (-1,1),` then roots of the quadratic equation `(a-1)x^2+a x+sqrt(1-a^2)=0` are a. real b. imaginary c. both equal d. none of these

Options

A. real
B. imaginary
C. both equal
D. none of these

Answer

1

Explanation

`(a - 1) x^(2) + ax + sqrt(1 - a^(2)) = 0`
`therefore D = a^(2) - 4(a - 1) sqrt(1 - a^(2))`
`= a^(2) - 4a sqrt(1 - a^(2)) + 4 sqrt(1 - a^(2))`
`= (a - 2 sqrt(1 - a^(2)))^(2) + 4sqrt(1 - a^(2))(1 - sqrt(1 - a^(2)))`
`ge 0` for `a in (-1, 1)`
Hence roots are real but not equal.