If $(x, y)$ is a variable point on the curve $x^2 + y^2 - 2x - 2y - 2 = 0$,then the minimum value of the expression $\frac{8}{(x - 1)^2} - \frac{(y - 1)^2}{4}$ is equal to

The equation of a common tangent to the circle $x^2+y^2=16$ and the ellipse $\frac{x^2}{49}+\frac{y^2}{4}=1$ is

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 $A$ be a point on the $x$-axis. Common tangents are drawn from $A$ to the curves $x^2+y^2=8$ and $y^2=16x$. If one of these tangents touches the two curves at $Q$ and $R$,then $(QR)^2$ is equal to

The focus of the parabola $y^2 = 4x + 16$ is the centre of the circle $C$ of radius $5$. If the values of $\lambda$,for which $C$ passes through the point of intersection of the lines $3x - y = 0$ and $x + \lambda y = 4$,are $\lambda_1$ and $\lambda_2$ (where $\lambda_1 < \lambda_2$),then $12\lambda_1 + 29\lambda_2$ is equal to . . . . . . .

Let a line perpendicular to the line $2x - y = 10$ touch the parabola $y^2 = 4(x - 9)$ at the point $P$. The distance of the point $P$ from the centre of the circle $x^2 + y^2 - 14x - 8y + 56 = 0$ is ...........