If the roots of the equation (a^2+b^2) x^2-2 b(a+c) x+(b^2+c^2)=0 are equal, then which one of the following is correct?

Correct answer: B. b^2=a c Since roots of the equation are equal then b^2-4 a c=0 i.e., 4 b^2(a+c)^2-4(a^2+b^2)(b^2+c^2)=0 b^2(a^2+2 a c+c^2)-(a^2 b^2+a^2 c^2+b^4+b^2 c^2)=0 a^2 b^2+2 a b^2 c+b^2 c^2-a^2 b^2-a^2 c^2-b^4-b^2 c^2=0 2 a b^2 c-a^2 c^2-b^4=0 b^4-2 a b^2 c+a^2 c^2=0 (b^2-a c)^2=0 b^2=a c .

NDA 2010

Exam: NDA
Year: 2010
Subject: Mathematics
Chapter: Quadratic Equation

NDA 2010 · Mathematics · Quadratic Equation

If the roots of the equation (a^2+b^2) x^2-2 b(a+c) x+(b^2+c^2)=0...

If the roots of the equation

\left(a^2+b^2\right) x^2-2 b(a+c) x+\left(b^2+c^2\right)=0

are equal, then which one of the following is correct?

Options

A $2 b=a+c$
B $b^2=a c$
C $b+c=2 a$
D $b=a c$

Correct answer

B. $b^2=a c$

Step-by-step solution

Since roots of the equation are equal then

b^2-4 a c=0

i.e.,

4 b^2(a+c)^2-4\left(a^2+b^2\right)\left(b^2+c^2\right)=0\Rightarrow b^2\left(a^2+2 a c+c^2\right)-\left(a^2 b^2+a^2 c^2+b^4+b^2 c^2\right)=0\Rightarrow a^2 b^2+2 a b^2 c+b^2 c^2-a^2 b^2-a^2 c^2-b^4-b^2 c^2=0\Rightarrow 2 a b^2 c-a^2 c^2-b^4=0\Rightarrow b^4-2 a b^2 c+a^2 c^2=0\Rightarrow\left(b^2-a c\right)^2=0\Rightarrow b^2=a c

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