A) \[(b+d)(ad+be)+{{(e-a)}^{2}}(a+c+e)=0\]
B) \[(b+d)(ad+be)+{{(e+a)}^{2}}(a+c+e)=0\]
C) \[(b-d)(ad-be)+{{(e-a)}^{2}}(a+c+e)=0\]
D) \[(b-d)(ad-be)+{{(e+a)}^{2}}(a+c+e)=0\]
Correct Answer: A
Solution :
Let \[a{{y}^{4}}+bx{{y}^{3}}+c{{x}^{2}}{{y}^{2}}+d{{x}^{3}}y+e{{x}^{4}}\] \[=(a{{x}^{2}}+pxy-a{{y}^{2}})({{x}^{2}}+qxy+{{y}^{2}})\] Comparing the coefficient of similar terms. We get, \[b=aq-p,\ c=-pq\], \[d=aq+p,\ e=-a\] \[b+d=2aq,\ e-a=-2a\] \[ad+be=2ap,a+c+e=-pq\] \[(b+d)(ad+be)=-{{(e-a)}^{2}}(a+c+e)\] \ \[(b+d)(ad+eb)+{{(e-a)}^{2}}(a+c+e)=0\].
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