2010006007
Level:
A
Consider the circle \(x^{2} + y^{2} +4x+ 6y + 10 = 0\).
Find the radius of the circle.
\(\sqrt{3}\)
\(2\)
\(3\)
\(9\)
Conics
2010006006
Level:
C
Find the tangent equations from the point \( M=[-2;2] \) to the parabola given by \( x^2+4x-4y+16=0 \).
\( x-y+4=0 \), \( x+y=0 \)
\( x+y-4=0 \), \( x+y=0 \)
\( x+y+4=0 \), \( x-y=0 \)
\( x-y-4=0 \), \( x-y=0 \)
\( x+y-4=0 \), \( x-y=0 \)
2010006005
Level:
C
Find the tangent equations from the point \( N=[0;0] \) to the ellipse given by \( 2x^2+y^2-4x-8y+12=0\).
\( 5x+y=0 \), \( x-y=0 \)
\( 5x-y=0 \), \( x+y=0 \)
\( 5x-y=0 \), \( x-y=0 \)
\( x+5y=0 \), \( x-y=0 \)
\( x-5y=0 \), \( x+y=0 \)
2010006004
Level:
C
The equation of a parabola is given by \( x^2 -8x +3y-2=0 \). Find the equation of a line, which passes through the vertex of this parabola and is parallel to the line \( 2x-5y+8=0 \).
\( -2x+5y-22 = 0 \)
\( 2x-5y-22 = 0 \)
\( 2x-5y-38 = 0 \)
\( 2x-5y+38 = 0 \)
\( -2x+5y+22 = 0 \)
2010006003
Level:
C
Find the set of values of the parameter \( c\in\mathbb{R} \) for which the line \( 5x-3y-c=0 \) isn't a secant of the ellipse \( 25x^2+16y^2=400 \).
\( (-\infty;-25 ] \cup [ 25;\infty) \)
\( (-25;25) \)
\( \{-25,25\} \)
\( (-\infty;-25) \cup (25;\infty) \)
\( [ 25;\infty) \)
2010006002
Level:
C
Find the value of the parameter \( p\in\mathbb{R} \) for which the line \( 2x-y-1=0 \) is a tangent of the parabola \( x^2=2py \).
\( \frac12 \)
\(- \frac12 \)
\( 2 \)
\( -2 \)
\( 1 \)
2010006001
Level:
C
Find the value of the parameter \( r\in\mathbb{R} \) for which the line \( 2x-y-1=0 \) is a tangent of the hyperbola \( 2x^2-y^2=r \).
\( 1 \)
\( -1 \)
\( 2 \)
\( -3 \)
\( 3 \)
2010005909
Level:
B
Which of the given points is one of the vertices of the hyperbola \(25y^{2} - 4x^{2} - 24x + 50y - 111 = 0\)?
\([-3;1]\)
\([3;1]\)
\([-3;4]\)
\([3;4]\)
2010005908
Level:
A
Which of the given points is one of the co-vertices on the minor axis of the ellipse \(25x^{2} + 9y^{2} + 100x - 54y - 44 = 0\)?
\([-5;3]\)
\([-2;0]\)
\([-2;-2]\)
\([3;1]\)
2010005907
Level:
B
Find the vertex of the following parabola.
\[
y^{2} + 12x - 6y - 15 = 0
\]
\([2;3]\)
\([-2;3]\)
\([2;-3]\)
\([-2;-3]\)