ba + a²c + ac2 = 3abc. If a,b are r...
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ba + a²c + ac2 = 3abc. If a,b are roots of the equation ax + bx + C = 0, 3. then find the value of laa + b)2 + Tab + b)2.

JEE/Engineering Exams
Maths
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3. ( quad a x^{2}+b x+c=0 ) [ begin{array}{l}alpha_{1} beta=r text { oots. } Rightarrow quad alpha+beta=-frac{b}{a} quad & quad alpha beta=frac{c}{a} text { Is } Rightarrow quad a alpha+a beta=-b quad text { & } alpha beta=frac{c}{a}end{array} ] ( Rightarrow quad a alpha+b=a beta ) ( & quad a beta+b=-a alpha ) ( begin{aligned} & frac{1}{(a alpha+b)^{2}}+frac{1}{(a beta+b)^{2}}=frac{1}{(-a beta)^{2}}+frac{1}{(-a alpha)^{2}} &=frac{1}{a^{2} beta^{2}}+frac{1}{a^{2} alpha^{2}} &=frac{1}{a^{2}}left(frac{1}{alpha^{2}}+frac{1}{beta^{2}}right)=frac{1}{a^{2}}left(frac{alpha^{2}+beta^{2}}{alpha^{2} beta^{2}}right) &=frac{1}{a^{2}}left(frac{(alpha+beta)^{2}-2 alpha beta}{(alpha beta)^{2}}right) &=frac{1}{a^{2}}left(frac{frac{b^{2}}{a^{2}}-frac{2 c}{a}}{frac{c^{2}}{a^{2}}}right)=frac{1}{a^{2}}left(frac{a b^{2}-2 a c}{c^{2}}right) =& frac{b^{2}-2 a c}{a^{2} c^{2}} end{aligned} )
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