Question

If $\cos ^4 \alpha+a, \sin ^4 \alpha+a$ are roots of the equation $x^2+2 k x+k=0$ and $\cos ^2 \alpha+b$, $\sin ^2 \alpha+b$ are roots of the equation $x^2+3 x+2=0$ then find the sum of all possible values of $k$.

Answer 1

Let $x_1=\cos ^4 \alpha+a, x_2=\sin ^4 \alpha+a, x_3=\cos ^2 \alpha+b$ and $x_4=\sin ^2 \alpha+b$
Now $x _1- x _2=\cos 2 \alpha$ and $x _3- x _4=\cos 2 \alpha$
Hence $\left|x_1-x_2\right|=\left|x_3-x_4\right|$
$\Rightarrow \left|x_1-x_2\right|^2=\left|x_3-x_4\right|^2$
$\Rightarrow \left(x_1+x_2\right)^2-4 x_1 x_2=\left(x_3+x_4\right)^2-4 x_3 x_4 $
$\Rightarrow 4 k^2-4 k=9-8=1 \Rightarrow 4 k^2-4 k-1=0$
Hence $k _1+ k _2=\frac{4}{4}=1$ Ans.
Note that given that roots of the equation $x^2+2 kx + k =0$ are $\left(\cos ^2 \alpha+ b \right)$ and $\left(\sin ^2 \alpha+ b \right)$ and difference is $\cos 2 \alpha=\sqrt{(5)^2-4 \cdot 3}=\sqrt{13}$ which is not possible, but if is wrong then is also wrong.