If the quadratic equation `a x^2+b x+c=0(a >0)` has `sec^2thetaa n dcos e c^2theta` as its roots, then which of the following must hold good? `b+c=0` b. `b^2-4a cgeq0` c. `ccgeq4a` d. `4a+bgeq0`

Options

A. ` b + c = 0`
B. ` b^(2) - 4 a ge 0 `
C. ` c ge 4a `
D. ` 4a b ge 0 `

Answer

1,2,3

Explanation

` sec^(2) theta + cosec^(2) theta = sec^(2) theta , cosec^(2) theta`
Sum of the roots is equal product and the roots are
real . Hence
` - (b)/(a) = (c)/(a)`
or ` b + c = 0`
Also ` b^(2) - 4ac ge 0`
or `c^(2) - 4ac ge 0 `
`or c(c - 4a) ge 0 `
` c - 4 a ge 0 (because c gt 0 )`
Futher
` b^(2) + 4ab ge 0 `
or `b + 4a le 0 (because b lt 0 )`