Conic Sections

9000123103

Level: 
C
The ellipse \[ 5x^{2} + 9y^{2} = 45 \] has tangent \(2x + 3y = 9\). Find the values of the real parameter \(k\) which ensure that the line \(y = kx + 3\) is a secant for the ellipse.
\(k\in \left (-\infty ,-\frac{2} {3}\right )\cup \left (\frac{2} {3},\infty \right )\)
\(k\in \left [ -\frac{2} {3}, \frac{2} {3}\right ] \)
\(k\in \left (-\frac{2} {3}, \frac{2} {3}\right )\)
\(k\in \left (-\infty ,-\frac{2} {3}\right ] \cup \left [ \frac{2} {3},\infty \right )\)

9000123107

Level: 
C
In the following list identify a line such that the line has a unique intersection with the hyperbola \[ x^{2} - y^{2} = 5 \] but the line is not the tangent to this hyperbola.
\(p\colon \frac{x} {5} + \frac{y} {5} = 1\)
\(p\colon y = 5x\)
\(p\colon 2x + y = 5\)
\(\begin{aligned}[t] p\colon x& = 1 & \\y & = -1 + t\text{, }t\in \mathbb{R} \\ \end{aligned}\)

9000117706

Level: 
C
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 \(R\).
\(x^{2} + (y + R)^{2} = (R + h)^{2}\)
\(x^{2} + y^{2} = (R + h)^{2}\)
\(x^{2} + (y + R)^{2} = h^{2}\)
\(x^{2} + y^{2} = h^{2}\)

9000120005

Level: 
B
The executives of a camp organize a holiday game. For this game it is important that the direct distance kitchen - tent - fireplace is equal for all tents in the camp. Is this information enough to determine the curve passing through all the tents in the camp? Is this curve a conic? If yes, determine which conic.
Yes, all the tents are on an ellipse.
Yes, all the tents are on a circle.
Yes, all the tents are on a parabola.
Yes, all the tents are on a hyperbola.
No, we do not have enough information to draw any conclusion.

9000117701

Level: 
C
A body is thrown at the initial angle \(\alpha = 30^{\circ }\) and the initial velocity \(v_{0} = 20\, \mathrm{m}/\mathrm{s}\). The trajectory of the body is a part of parabola. Find the directrix of this parabola. Hint: The coordinates of the moving body as functions of time are \[ \begin{aligned}x& = v_{0}t\cdot \cos \alpha , & \\y& = v_{0}t\cdot \sin \alpha -\frac{1} {2}gt^{2}. \\ \end{aligned} \] Consider the standard acceleration due to gravity \(g = 10\, \mathrm{m}/\mathrm{s}^{2}\).
\(y = 20\)
\(y = 5\)
\(y = 15\)
\(y = 10\)

9000117702

Level: 
C
The Earth travels around the Sun on an elliptical orbit. The Sun is in the focus of this ellipse. The maximal distance from Earth to the Sun is \(152.1\cdot 10^{6}\, \mathrm{km}\), the minimal distance from Earth to the Sun is \(147.1\cdot 10^{6}\, \mathrm{km}\). Find the length of the semi-minor axis (one half of the length of the shorter axis) and round your answer to the nearest \(10^{4}\, \mathrm{km}\).
\(149.58\cdot 10^{6}\, \mathrm{km}\)
\(2.58\cdot 10^{6}\, \mathrm{km}\)
\(299.21\cdot 10^{6}\, \mathrm{km}\)
\(149.61\cdot 10^{6}\, \mathrm{km}\)

9000117703

Level: 
C
For an isothermal process in an ideal gas the product \(pV \) is constant (Boyle's law). In a pressure-volume diagram which shows \(p\) as a function of \(V \) this law describes a hyperbola (called isotherm). Do we have enough information to identify the asymptotes? If so, find these asymptotes.
\(p = 0\), \(V = 0\)
\(p = V \), \(p = -V \)
\(p = 0\), \(p = V \)
It is not possible to draw any conclusion.