Assertion and Reason Question on Algebraic Identities
Each of the following questions contains STATEMENT-1 (Assertion) and STATEMENT-2 (Reason) and has following four choices (a), (b), (c) and (d), only one of which is the correct answer.
(a) Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.
(b) Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.
(c) Statement-1 is true, Statement-2 is false.
(d) Statement-1 is false, Statement-2 is true.
Statement-1 (Assertion):
If \(a,b,c\) are all non-zero such that \[ a+b+c=0, \] then \[ \frac{a^2}{bc}+\frac{b^2}{ca}+\frac{c^2}{ab}=3 \]
Statement-2 (Reason):
If \[ a+b+c=9 \] and \[ a^2+b^2+c^2=35, \] then \[ ab+bc+ca=23 \]
Solution
For Statement-1:
\[ \frac{a^2}{bc}+\frac{b^2}{ca}+\frac{c^2}{ab} = \frac{a^3+b^3+c^3}{abc} \]
Since \[ a+b+c=0, \] using identity \[ a^3+b^3+c^3=3abc \]
\[ \frac{a^3+b^3+c^3}{abc} = \frac{3abc}{abc} =3 \]
Hence, Statement-1 is true.
For Statement-2:
Using identity:
\[ a^2+b^2+c^2 = (a+b+c)^2-2(ab+bc+ca) \]
\[ 35=9^2-2(ab+bc+ca) \]
\[ 35=81-2(ab+bc+ca) \]
\[ 2(ab+bc+ca)=46 \]
\[ ab+bc+ca=23 \]
Hence, Statement-2 is also true.
But Statement-2 is not the correct explanation for Statement-1.
\[ \boxed{(b)} \]