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.
\(p\colon \frac{x} {5} + \frac{y} {5} = 1\)
\(p\colon y = 5x\)
\(p\colon 2x + y = 5\)
\(\begin{aligned}[t] p\colon x& = 1 & \\y & = -1 + t\text{, }t\in \mathbb{R} \\ \end{aligned}\)

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.

9000120007

Level: 
B
On a map of a city, the town hall is represented by a point and a river through the city by a straight line. There are places in the city with the property that the direct distance from each place to the town hall is equal to the direct distance to the river. In the following list identify a curve which can be used to join all these places.
parabola
circle
ellipse
hyperbola
none of them

9000117704

Level: 
C
Given physical quantities and laws relating these quantities, identify an answer where the graph which relates these quantities is a part of a hyperbola. (The other quantities are supposed to be constant.)
The pressure (\(p\)) and the area (\(S\)) over which the pressure is distributed, if \(F = p\cdot S\).
The mass (\(m\)) and the kinetic energy (\(E_{k}\)) of a moving body, if \(E_{k} = \frac{1} {2}\cdot m\cdot v^{2}\).
The velocity (\(v\)) and the kinetic energy (\(E_{k}\)) of a moving body, if \(E_{k} = \frac{1} {2}\cdot m\cdot v^{2}\).
The mass (\(m\)) and the potential energy (\(E_{p}\)) in a homogeneous gravitational field, if \(E_{p} = m\cdot g\cdot h\).

9000117705

Level: 
C
Given physical quantities and laws relating these quantities, identify an answer where the graph which relates these quantities is a part of a parabola. (The other quantities are supposed to be constant.)
The electrical work (\(W\)) and the current (\(I\)), if \(W = R\cdot I^{2}\cdot t\).
The mass (\(m\)) and the acceleration (\(a\)) of a moving body, if \(F = m\cdot a\).
The height (\(h\)) and the potential energy (\(E_{p}\)), if \(E_{p} = m\cdot g\cdot h\).
The electrical work (\(W\)) and the time (\(t\)), if \(W = R\cdot I^{2}\cdot t\).