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. non-real complex
B. real and equal
C. integers
D. real and disinct

Answer

4

Explanation

Given that ` a lt b lt c lt d ` . Let
` f(x) = (x - a) (x - c) + 2(x - b) (x - d)`
`rArr f(b) = (b - a) (b - c) lt 0 `
and ` f(d) = (d - a) (d - c) (d - c) gt 0 `
Hence , f(x) = 0 has one root in (b, d) , Also `f(a) f(c) lt 0 `. So the
other root lies in (a,c) . Hence, roots of the equation are real
and disinct .