A) \[{{a}^{2}}+{{c}^{2}}=0\]
B) \[{{a}^{2}}+ac+bd+{{d}^{2}}=0\]
C) \[{{a}^{2}}{{c}^{2}}+bd+{{d}^{2}}=0\]
D) None of these
Correct Answer: B
Solution :
Let y = mx be any line represented by the equation \[a{{x}^{3}}+b{{x}^{2}}y+cx{{y}^{2}}+d{{y}^{3}}=0\] \[\Rightarrow \]\[a{{x}^{3}}+b{{x}^{2}}(mx)+x({{m}^{2}}{{x}^{2}})+d{{m}^{3}}{{x}^{3}}=0\] \[\Rightarrow \]\[a+bm+c{{m}^{2}}+d{{m}^{3}}=0\], which is a cubic equation. It represents three lines out of which two are perpendicular hence \[{{m}_{1}}{{m}_{2}}=-1\] and\[{{m}_{1}}{{m}_{2}}{{m}_{3}}=-\frac{a}{d}\Rightarrow {{m}_{3}}=\frac{a}{d}\]and\[{{m}_{3}}\]is the root of the given equation Hence, \[a+b\left( \frac{a}{d} \right)+c{{\left( \frac{a}{d} \right)}^{2}}+d{{\left( \frac{a}{d} \right)}^{3}}=0\] \[\Rightarrow \]\[{{d}^{2}}+bd+ca+{{a}^{2}}=0\]
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