If $\alpha, \beta$ are roots of the equation $x^{2}+5 \sqrt{2} x+10=0$,$\alpha > \beta$ and $P_{n}=\alpha^{n}-\beta^{n}$ for each positive integer $n$,then the value of $\left(\frac{P_{17} P_{20}+5 \sqrt{2} P_{17} P_{19}}{P_{18} P_{19}+5 \sqrt{2} P_{18}^{2}}\right)$ is equal to $....$

If $\alpha$ and $\beta$ are the real roots of the equation $x^2+ax+b=0$,where $\alpha+\beta=\frac{1}{2}$ and $\alpha^3+\beta^3=\frac{37}{8}$,then find the value of $a-\frac{1}{b}$.

If the ratio of the roots of the equation $px^2+qx+r=0$ is $a:b$,then $\frac{ab}{(a+b)^2}=$

If the product of the roots of the equation $x^2 - 3kx + 2e^{2\log k} - 1 = 0$ is $7$,then find the value of $k$.

Let $a, b, c$ be in arithmetic progression. Let the centroid of the triangle with vertices $(a, c), (2, b)$ and $(a, b)$ be $\left(\frac{10}{3}, \frac{7}{3}\right)$. If $\alpha, \beta$ are the roots of the equation $ax^{2} + bx + 1 = 0$,then the value of $\alpha^{2} + \beta^{2} - \alpha\beta$ is ....... .

Let $x, y$ be real numbers such that $x \neq y$ and $xy \neq 1$. If $ax + b \sec(\tan^{-1} x) = c$ and $ay + b \sec(\tan^{-1} y) = c$,then $\frac{x+y}{1-xy} =$