If the left hand side of the equation
`a(b-c)x^2+b(c-a) xy+c(a-b)y^2=0` is a perfect square , the value of
`{(log(a+c)+log(a-2b+c)^2)/log(a-c)}^2`, `(a,b,cinR^+,agtc)` is

Options

A. 3
B. 4
C. 2
D. 1

Answer

C

Explanation

If `a(b-c)x^(2)| b(c " "a) xy| c(a-b) y^(2) = 0` is n perfect square, then
`b^(2)(c-a)^(2)y^(2)-4ac (b-c)(a-b) y^(2) = 0`
`rArr b^(2)(c-a)^(2)-4ac(b-c)(a-b) = 0`
`rArr b^(2) (c+a)^(2) = 4ac (ab-ac+bc)`
`rArr{b(c+a)"^(2)-4abc (a+c) + 4a^(2)c^(2) =0`
`rArr {b(c+a)-2ac}^(2) = 0 rArr b = (2ac)/(a+c)`
Now,
`("log"(a+c) + "log"(a-2b+c))/("log"(a-c))`
`("log"(a+c) + "log"(a-(4ac)/(a+c)+c))/("log"(a-c)) = ("log"(a-c)^(2))/("log"(a-c)) = 2`