If `a ,b ,c ,d` are four consecutive terms of an increasing A.P., then the roots of the equation `(x-a)(x-c)+2(x-b)(x-d)=0` are a. non-real complex b. real and equal c. integers d. real and distinct

Options

A. real and distinct
B. non-real complex
C. real and equal
D. integers

Answer

A

Explanation

Let `lambda` be the common difference of the increasing A.P. Then, `b=a+lambda, c = a + 2 lambda and d = a + 3 lambda, "where" lambda gt 0`.
`therefore" "(x-a)(x-c)+2(x-b)(x-d)=0`
`rArr" "3x^(2)-2(3a+5 lambda)x+a(a+ 2 lambda)+2(a+lambda)(a+3 lambda)=0`
Let D be its discriminant. Then, `D = 4(3a+5 lambda)^(2)-12a(a+2 lambda)-24(a+lambda)(a+3 lambda)`
`rArr" "D = 28 lambda^(2) gt 0`.
Hence, the roots of the given equation are real and distinct.