9000151309
Level:
B
Given points \(A = [-1.4]\),
\(B = [2,-2]\),
\(C = [5,-1]\), find the angle
\(\varphi \) between
the lines \(AB\)
and \(BC\).
\(81^{\circ }52'\)
\(98^{\circ }08'\)
\(61^{\circ }22'\)
\(45^{\circ }32'\)
Analytical Plane Geometry
9000151310
Level:
A
Find the value of the real parameter
\(a\) which ensures, that
the following two lines \(p\)
and \(q\)
are perpendicular.
\[
p\colon ax + y - 4 = 0,\qquad q\colon x + 2y + 4 = 0.
\]
\(- 2\)
\(2\)
\(1\)
\(- 1\)
9000151303
Level:
B
Find the angle \(\varphi \)
between the lines given by their slope-intercept equations
\(y = 6\) and
\(y = \frac{3}
{4}x\).
\(36^{\circ }52'\)
\(45^{\circ }59'\)
\(64^{\circ }33'\)
\(76^{\circ }11'\)
9000151305
Level:
B
Find the angle \(\varphi \)
between the lines \(x + y + 1 = 0\)
and \(x - y - 1 = 0\).
\(90^{\circ }\)
\(0^{\circ }\)
\(30^{\circ }\)
\(60^{\circ }\)
9000151301
Level:
B
Find the angle \(\varphi \)
between the lines \(3x - 7 = 0\)
and \(x + y + 13 = 0\).
\(45^{\circ }\)
\(90^{\circ }\)
\(60^{\circ }\)
\(30^{\circ }\)
9000151302
Level:
B
Find the angle \(\varphi \)
between the lines \(p\)
and \(q\)
given by their parametric equations.
\[
p\colon \begin{aligned}[t] x& = 1 + 2t, &
\\y& = 3 - 3t,\ t\in \mathbb{R},
\\ \end{aligned}\qquad q\colon \begin{aligned}[t] x& = 2 - k, &
\\y& = 3 + k,\ k\in \mathbb{R}
\\ \end{aligned}
\]
\(11^{\circ }19'\)
\(88^{\circ }41'\)
\(45^{\circ }45'\)
\(54^{\circ }12'\)
9000151306
Level:
B
Find the angle \(\varphi \)
between the lines \(p\)
and \(q\)
given by their parametric equations.
\[
p\colon \begin{aligned}[t] x& = 1 - t, &
\\y& = 2 + t,\ t\in \mathbb{R},
\\ \end{aligned}\qquad q\colon \begin{aligned}[t] x& = 4 - k, &
\\y& = 5 + k,\ k\in \mathbb{R}.
\\ \end{aligned}
\]
\(0^{\circ }\)
\(90^{\circ }\)
\(60^{\circ }\)
\(30^{\circ }\)
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
\(\sqrt{10}\).
\(p_{1}\colon x - 3y + 12 = 0\),
\(p_{2}\colon x - 3y - 8 = 0\)
\(p\colon x - 3y = 0\)
\(p\colon x - 3y + \sqrt{10} = 0\)
\(p_{1}\colon x - 3y + \sqrt{10} = 0\),
\(p_{2}\colon x - 3y -\sqrt{10} = 0\)
9000149401
Level:
B
Find the distance from the point \(P = [-4,2]\)
to the line \(3x - 4y - 5 = 0\).
\(5\)
\(1\)
The line contains \(P\).
\(\sqrt{5}\)
9000149403
Level:
B
Find the distance from the point \(M = [1,1]\)
to the line \(p\).
\[
\begin{aligned}p\colon x& = 3 + t, &
\\y & = 1 + t,\ t\in \mathbb{R}
\\ \end{aligned}
\]
\(\sqrt{2}\)
\(2\)
\(1\)
\(0\) (the point is
on the line \(p\) )