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

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 \(W = R\cdot I^{2}\cdot t\).

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 \(10^{4}\, \mathrm{km}\).
\(149.58\cdot 10^{6}\, \mathrm{km}\)
\(2.58\cdot 10^{6}\, \mathrm{km}\)
\(299.21\cdot 10^{6}\, \mathrm{km}\)
\(149.61\cdot 10^{6}\, \mathrm{km}\)