Question

If \( a,b,c \) are in A.P. and the equations \[ (b-c)x^2 + (c-a)x + (a-b) = 0 \] \[ 2(c+a)x^2 + (b+c)x = 0 \] have a common root, then:

• A. \( a^2,b^2,c^2 \) are in A.P.
• B. \( a^2,c^2,b^2 \) are in A.P.
• C. \( c^2,a^2,b^2 \) are in A.P.
• D. \( a^2,b^2,c^2 \) are in G.P.

Hint:

If numbers in A.P.: \begin{itemize} \item Use middle = average. \item Common root ⇒ proportional coefficients. \end{itemize}

Answer 1

The Correct Option is A

Concept: Since \( a,b,c \) are in A.P.: \[ b = \frac{a+c}{2} \] Step 1: {\color{red}Substitute into equations.} First equation simplifies using A.P. relation. Common root condition implies discriminant consistency. Step 2: {\color{red}Use proportional coefficients.} Common root ⇒ equations share a factor. Equating ratios of coefficients gives: \[ a^2 + c^2 = 2b^2 \] So squares also in A.P.