`If a, b, c, d in R` such that `a lt b lt c lt d,` then roots of the equation `(x-a)(x-c)+2(x=b)(x-d) = 0`

Options

A. are imaginary
B. are equal
C. are real and distinct lying between a and b
D. real and distinct lying between a and d.

Answer

D

Explanation

Let `f(x) = (x-a) (x-c) + 2(x-b) (x-d)`. Then,
`f(a)=2(a-b)(a-d) gt 0" "[therefore a-b lt 0 and a-d lt 0]`
`f(b)=(b-a)(b-c) lt 0" "[therefore b-a gt 0 and b-c lt 0]`
and, `f(d)=(d-a)(d-b) gt 0" "[therefore d-a gt 0 and d-b gt 0]`
Thus, one root of f(x) = 0 lies between a and b and another root lies between b and d. Hence, the roots of the given equation are real and distinct and between a and d.