C

Find all the tangents to the hyperbola $x^2-2y^2=8$ such that the angle between each tangent and the $x$-axis is ${45}^{\circ }$.

A force due to gravity on a body is $1800\mathrm{N}$. This body has to be lifted to the height $50\mathrm{cm}$ using a slope. The maximal force which can be used to lift the body is $600\mathrm{N}$. Neglect the friction and find the minimal length of the slope required to accomplish this task.Hint: The force due to gravity can be decomposed into two directions. The normal force $F_1$ is compensated by the reaction of the slope. The force $F_2$ parallel to the slope is required to overcome if we wish to lift the body (see the picture).

Satellites travel along approximately circular paths. Consider a satellite in the height $h$ measured from the Earth surface. Further, consider the coordinate system with origin on the Earth surface directly below the satellite and the $y$-axis oriented up (to the satellite). The $x$-axis is perpendicular to $y$-axis and it is in the plane defined by the trajectory of the satellite. Neglect the Earth's rotation and find the equation which describes the path of the satellite. The Earth radius is $R$.

The side of a regular hexagonal prism $ABCDEF{A}^{\prime }{B}^{\prime }{C}^{\prime }{D}^{\prime }{E}^{\prime }{F}^{\prime }$ is $a=3\mathrm{cm}$ and the height $v=8\mathrm{cm}$. Find the length of the diagonal $A{D}^{\prime }$.