9000105402
Level:
B
The parabola \(P\colon x^{2} - 4x - 10y - 21 = 0\) has
two \(x\)-intercepts.
Find their distance.
\(10\)
\(12\)
\(8\)
\(6\)
Conics
9000105403
Level:
B
The parabola \(P\colon y^{2} + 2y + 10x - 24 = 0\) has
two \(y\)-intercepts.
Find their distance.
\(10\)
\(8\)
\(6\)
\(4\)
9000105404
Level:
B
The parabola \(P\colon y^{2} + 4y + 6x - 5 = 0\) has
two \(y\)-intercepts.
Find their distance.
\(6\)
\(8\)
\(10\)
\(4\)
9000105405
Level:
B
Parabola is a set of the points that are equidistant from the point (focus)
and the line (directrix). Find the equation of the directrix of the parabola
\(P\colon x^{2} - 4x - 6y - 17 = 0\).
\(y + 5 = 0\)
\(y - 3 = 0\)
\(x + 4 = 0\)
\(x - 2 = 0\)
9000105406
Level:
B
Parabola is a set of the points that are equidistant from the point (focus)
and the line (directrix). Find the equation of the directrix of the parabola
\(P\colon y^{2} + 4y + 4x - 4 = 0\).
\(x - 3 = 0\)
\(x + 4 = 0\)
\(y + 1 = 0\)
\(y - 2 = 0\)
9000105408
Level:
B
Parabola is a set of the points that are equidistant from the point (focus)
and the line (directrix). Find the equation of the directrix of the parabola
\(P\colon x^{2} - 8x + 6y + 19 = 0\).
\(y - 1 = 0\)
\(y + 2 = 0\)
\(x + 4 = 0\)
\(x - 3 = 0\)
9000105407
Level:
B
Parabola is a set of points that are equidistant from the point (focus)
and the line (directrix). Find the equation of the directrix of the parabola
\(y^{2} + 6y - 12x + 21 = 0\).
\(x + 2 = 0\)
\(x - 5 = 0\)
\(y + 3 = 0\)
\(y - 1 = 0\)
9000105409
Level:
B
Find the distance between the point \([7;3]\)
and the focus of the parabola \(x^{2} - 8x + 8y + 8 = 0\).
\(5\)
\(9\sqrt5\)
\(3\)
\(\sqrt{13}\)
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.