9000149404
Level:
B
Given points \(A = [-3;13]\),
\(K = [0;4]\),
\(L = [-5;-6]\), find the distance
from the point \(A\)
to the line \(KL\).
\(3\sqrt{5}\)
\(3\)
\(5\)
\(\sqrt{5}\)
B
9000149407
Level:
B
Find the distance between the lines \(p\colon 3x - 4y + 1 = 0\)
and \(q\colon 3x - 4y + 4 = 0\).
\(\frac{3}
{5}\)
\(1\)
\(4\)
\(0\) (the
lines have an intersection)
9000149410
Level:
B
Find all lines passing through the point
\(A = [-2;-6]\) such that the distance
from the point \([0.0]\)
to these lines is \(2\sqrt{2}\).
\(p_{1}\colon 7x + y + 20 = 0\),
\(p_{2}\colon x - y - 4 = 0\)
\(p\colon 7x - y = 0\)
\(p\colon x + y + 2\sqrt{2} = 0\)
\(p_{1}\colon x - y + 2\sqrt{2} = 0\),
\(p_{2}\colon x + y - 2\sqrt{2} = 0\)
9000149706
Level:
B
Find the center of the following hyperbola.
\[
4x^{2} - 3y^{2} + 8x - 30y - 49 = 0
\]
\([-1;-5]\)
\([-1;5]\)
\([1;-5]\)
\([1;5]\)
9000149707
Level:
B
Find the center of the following hyperbola.
\[
5x^{2} - 6y^{2} - 30x + 12y + 9 = 0
\]
\([3;1]\)
\([3;-1]\)
\([-3;1]\)
\([-3;-1]\)
9000149710
Level:
B
Find the vertex of the following parabola.
\[
x^{2} - 6x - 12y - 3 = 0
\]
\([3;-1]\)
\([3;1]\)
\([-3;1]\)
\([-3;-1]\)
9000149709
Level:
B
Find the vertex of the following parabola.
\[
y^{2} - 12x + 4y + 64 = 0
\]
\([5;-2]\)
\([5;2]\)
\([-5;2]\)
\([-5;-2]\)
9000149305
Level:
B
Given a translation \(T\)
of a plane, find the lines which are mapped to the same line by
\(T\).
All lines parallel to the translation vector are mapped into itself.
All lines perpendicular to the translation vector are mapped into itself.
There are no lines which are mapped into itself by the translation.
Every line is mapped into itself by the translation.
9000149409
Level:
B
Find all lines which are parallel to \(p\colon x - 3y + 2 = 0\)
and the distance from every of these lines to
\(p\) is
\(\sqrt{10}\).
\(p_{1}\colon x - 3y + 12 = 0\),
\(p_{2}\colon x - 3y - 8 = 0\)
\(p\colon x - 3y = 0\)
\(p\colon x - 3y + \sqrt{10} = 0\)
\(p_{1}\colon x - 3y + \sqrt{10} = 0\),
\(p_{2}\colon x - 3y -\sqrt{10} = 0\)
9000149408
Level:
B
On the \(x\)-axis
find the points such that the distance from these points to the line
\(p\colon x - 2y + 2 = 0\) is
\(\sqrt{5}\).
\([3;0]\),
\([-7;0]\)
\([5;0]\)
\(\left [\sqrt{5};0\right ]\),
\(\left [-\sqrt{5};0\right ]\)
\([3;7]\)