Conic Sections

9000123102

Level: 
C
Find a true statement about the ellipse \[ x^{2} + 4y^{2} - 8y = 0. \]
The tangent to the ellipse can pass through any point on the line \(y = -1\).
The tangent to the ellipse can pass through any point on the line \(x = 1\).
The tangent to the ellipse can pass through the point \([-1;1]\).
The tangent to the ellipse can pass through any point on the line \(y = 1\).

9000123106

Level: 
C
Find the tangent line \(q\) to the parabola \(4(y - 2) = (x + 1)^{2}\), so that the tangent \(q\) is parallel to the line \(p\colon 4x - 5y + 17 = 0.\)
\(q\colon 20x - 25y + 54 = 0\)
\(q\colon 20x - 25y - 27 = 0\)
\(q\colon 4x - 5y + 27 = 0\)
\(q\colon 4x -5y - 17 = 0\)

9000123108

Level: 
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 \(45^{\circ }\).
\(y = x + 2\text{, }y = x - 2\text{, }y = -x + 2\text{, }y = -x - 2\)
\(y = x + 2\text{, }y = x - 2\)
\(y = x + 2\text{, }y = -x + 2\)
\(y = x + 2\)

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.

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 \(E_{p} = m\cdot g\cdot h\).