Quadratic Identity

Q. Let $\alpha ,\beta ,\gamma$ be distinct real numbers such that
$a{\alpha }^2+b\alpha +c=\left(\sin \theta \right){\alpha }^2+\left(\cos \theta \right)\alpha a{\beta }^2+b\beta +c=\left(\sin \theta \right){\beta }^2+\left(\cos \theta \right)\beta a{\gamma }^2+b\gamma +c=\left(\sin \theta \right){\gamma }^2+\left(\cos \theta \right)\gamma$
where $a,b,c,\theta \in R$
Then which of the following is/are correct?

1. The maximum value of $\frac{a^2+b^2}{a^2+3ab+5b^2}$ is $2$
2. The minimum value of $\frac{a^2+b^2}{a^2+3ab+5b^2}$ is $\frac{2}{11}$
3. If $\overrightarrow{V_1}=a\hat{i}+b\hat{j}+c\hat{k}$ makes an angle $\frac{\pi }{3}$ with $\overrightarrow{V_2}=\hat{i}+\hat{j}+\sqrt{2}\hat{k}$, then the number of values of $\theta \in [0,2\pi ]$ is $3$.
4. If $\overrightarrow{V_1}=a\hat{i}+b\hat{j}+c\hat{k}$ makes an angle $\frac{\pi }{3}$ with $\overrightarrow{V_2}=\hat{i}+\hat{j}+\sqrt{2}\hat{k}$, then the number of values of $\theta \in [0,2\pi ]$ is $5$.

Q. For the polynomial equation
$\frac{(x+q)(x+r)}{(q-p)(r-p)}+\frac{(x+r)(x+p)}{(r-q)(p-q)}+\frac{(x+p)(x+q)}{(p-r)(q-r)}-1=0$, where $p,q,r$ are distinct real numbers.
Which of the following statement is correct?

1. It has $2$ distinct real roots.
2. It has no real roots.
3. It has real and equal roots.
4. It has more than $2$ real roots.