9000105409
Level:
B
Find the distance between the point \([7;3]\)
and the focus of the parabola \(x^{2} - 8x + 8y + 8 = 0\).
\(5\)
\(9\sqrt5\)
\(3\)
\(\sqrt{13}\)
Conics
9000104801
Level:
C
Consider the hyperbola
\[
xy = -1
\]
and a line \(p\)
parallel to one of the axes but not identical with this axis. Find the true statement.
The line \(p\)
has a unique intersection with the hyperbola.
The line \(p\)
has two intersections with the hyperbola.
The line \(p\)
does not have any intersection with the hyperbola.
We cannot draw any conclusion on the number of intersections of the line
\(p\) with
the hyperbola.
9000104803
Level:
C
Consider the hyperbola
\[
\frac{x^{2}}
{16} -\frac{y^{2}}
{4} = 1
\]
and a line \(p\)
parallel to one of the axes. Find the true statement.
We cannot draw any conclusion on the number of intersections of the line
\(p\) with
the hyperbola.
The line \(p\)
has two intersections with the hyperbola.
The line \(p\)
has a unique intersection with the hyperbola.
The line \(p\)
does not have any intersection with the hyperbola.
9000104805
Level:
C
Find the slope of a line through the center of the hyperbola
\[
\frac{(x - 2)^{2}}
{4} -\frac{(y + 3)^{2}}
{9} = 1
\]
which has a unique intersection with this hyperbola.
There is no solution, the line with these properties does not exist.
\(\frac{3}
{2}\)
\(-\frac{3}
{2}\)
\(\frac{2}
{3}\)
\(1\)
\(0\)
9000104809
Level:
C
Among the following lines (which all pass through the point
\([-1;3]\))
identify a line which is tangent to the following hyperbola.
\[
(x + 2)\cdot (y - 2) = 1
\]
\(k\colon \ y = -x + 2\)
\(p\colon \ y = 3\)
\(q\colon \ x = -1\)
\(r\colon \ y = x + 4\)
None of the given answers is correct.
9000097007
Level:
B
Parabola is a set of the points that are equidistant from the point
(focus) and the line (directrix). Find the directrix of the parabola
\((y - 4)^{2} = 8(x - 1)\).
\(x = -1\)
\(x = 4\)
\(y = 4\)
\(y = 2\)
9000097010
Level:
B
Parabola is a set of the points that are equidistant from the point
(focus) and the line (directrix). Find the directrix of the parabola
\((y + 3)^{2} = -8(x + 4)\).
\(x = -2\)
\(x = -4\)
\(y = -3\)
\(y = -2\)
9000091201
Level:
A
Consider the circle \(k\colon x^{2} - 4x + y^{2} + 6y + 12 = 0\).
Find the radius of the circle.
\(1\)
\(2\)
\(3\)
\(4\)
9000091202
Level:
A
Consider the circle \(k\colon x^{2} + 6x + y^{2} + 2y + 6 = 0\).
Find the radius of the circle.
\(2\)
\(\sqrt{2}\)
\(3\)
\(4\)
9000091204
Level:
A
Consider the circle \(k\colon x^{2} + 4x + y^{2} - 8y + 11 = 0\).
Find the radius of the circle.
\(3\)
\(1\)
\(2\)
\(4\)