For different real non-zero numbers $x_1, x_2, x_3$ and $x_4$,suppose the points $(x_1, \frac{1}{x_1}), (x_2, \frac{1}{x_2}), (x_3, \frac{1}{x_3})$ and $(x_4, \frac{1}{x_4})$ lie on the boundary of a circle of radius $4$. Then,the value of $x_1 x_2 x_3 x_4$ is

Let $C$ be the circle $x^2+(y-1)^2=2$. Let $E_1$ and $E_2$ be two ellipses whose centers lie at the origin and whose major axes lie on the $x$-axis and $y$-axis,respectively. Let the straight line $x+y=3$ touch the curves $C$,$E_1$,and $E_2$ at $P(x_1, y_1)$,$Q(x_2, y_2)$,and $R(x_3, y_3)$,respectively. Given that $P$ is the midpoint of the line segment $QR$ and $PQ = \frac{2\sqrt{2}}{3}$,the value of $9(x_1y_1 + x_2y_2 + x_3y_3)$ is equal to . . . . . . .

The tangent at $A(-1, 2)$ on the circle $x^2+y^2-4x-8y+7=0$ touches the circle $x^2+y^2+4x+6y=0$ at $B$. Then,a point of trisection of $AB$ is

Two chords of the circle $x^2+y^2-2gx-2hy+g^2+h^2-c^2=0$ pass through the point $(g, h+c)$,and the line $y=x$ bisects these two chords. Then:

$A$ pair of tangents are drawn from the origin to the circle $x^2 + y^2 + 20(x + y) + 20 = 0$. The equation of the pair of tangents is

The given circle $x^2 + y^2 + 2px = 0$,$p \in R$ touches the parabola $y^2 = 4x$ externally,then