Two sides $^{'} a^{'}$ and $^{'} b^{'}$ of triangle $A B C$ are given by the roots of the equation $x^{2} - 4 x + 1 = 0$ and the included angle between them is $\frac{π}{3}$ , then the value $(\frac{a + b}{c^{2}})$ is

\frac{1}{4}

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\frac{4}{13}

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\frac{4}{11}

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Solution

The correct option is C $\frac{4}{11}$

$x^{2} - 4 x + 1 = 0$
$[x - (2 - \sqrt{3})] [x - (2 + \sqrt{3})] = 0$
$C = \sqrt{a^{2} + b^{2} - 2 a b c o s A}$
$= \sqrt{2 + 3 - 4 \sqrt{3} + 2 + 3 + 4 \sqrt{3} + 2 (4 - 3) \frac{1}{2}}$
$= \sqrt{10 + 1} = \sqrt{11}$
$\Rightarrow \frac{a + b}{C^{2}} = \frac{2 + \sqrt{3} + 2 - \sqrt{3}}{11} = \frac{4}{11}$ [C]

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Two sides ′a′ and ′b′ of triangle ABC are given by the roots of the equation x2−4x+1=0 and the included angle between them is π3 , then the value (a+bc2) is

The correct option is C 411x2−4x+1=0[x−(2−√3)][x−(2+√3)]=0C=√a2+b2−2abcosA =√2+3−4√3+2+3+4√3+2(4−3)12 =√10+1=√11 ⇒a+bC2=2+√3+2−√311=411 [C]

Two sides $^{'} a^{'}$ and $^{'} b^{'}$ of triangle $A B C$ are given by the roots of the equation $x^{2} - 4 x + 1 = 0$ and the included angle between them is $\frac{π}{3}$ , then the value $(\frac{a + b}{c^{2}})$ is