Question

I: If $a,b,c$ are real, the roots of $(b-c)x^2+(c-a)x+(a-b)=0$ are real and equal, then $a,b,c$ are in A.P.
II: If $a,b,c$ are real and the roots of $(a^2+b^2)x^2-2b(a+c)x+b^2+c^2=0$ are real and equal, then $a,b,c$ are in H.P.
Which of the above statement(s) is(are) true?

• A. only I
• B. only II
• C. both I and II
• D. neither I or II

Answer 1

The correct option is A only I

Roots of $(b-c)x^2+(c-a)x+(a-b)=0$ are equal.
$\Rightarrow (c-a{)}^2=4(a-b\left)\right(b-c)$
$c^2+a^2-2ac=4ab-4b^2-4ac+4bc$
$(c+a{)}^2=4(ab+bc-b^2)$
$(\frac{c+a}{2}{)}^2=b(a+c-b)$
$a+c=2b\Rightarrow$
$\therefore$ a,b,c are in A.P.
roots of $(a^2+b^2)x^2-2b(a+c)x+b^2+c^2=0$ are equal
$\Rightarrow 4b^2(a+c{)}^2=4(a^2+b^2\left)\right(b^2+c^2)$
$b^2(a+c{)}^2=(a^2+b^2\left)\right(b^2+c^2)$
$b^2{a}^2+b^2{c}^2+2ac{b}^2=a^2{b}^2+b^4+a^2{c}^2+b^2{c}^2$
$2a{b}^{2}c=b^4+a^2{c}^2$
$(b^2-ac{)}^2=0$
$\therefore b^2=ac$. So $a,b,c$ are in GP