9000168710
Level:
B
Find the vertex of the hyperbola \(9x^{2} - 25y^{2} + 18x + 100y - 316 = 0\).
\([4;2]\)
\([2;2]\)
\([-1;-3]\)
\([-1;-1]\)
Conics
1003024103
Level:
C
Find the equation of the circle, which passes through the points \( A=[-3;2] \), \( B=[-1;4] \) and \( C=[3;0] \).
\( x^2 + (y-1)^2 = 10 \)
\( (x-1)^2 + y^2 = 10 \)
\( x^2 + (y-1)^2 = \sqrt{10} \)
\( x^2 + y^2 = 10 \)
\( x^2 + (y+1)^2 = 10 \)
1003024104
Level:
C
Find the points of intersection of the circle \( x^2+y^2=4 \) and the line \( x+y-2=0 \).
\( [0;2] \), \( [2;0] \)
\( [0;-2] \), \( [-2;0] \)
\( [0;-2] \), \( [2;0] \)
\( [0;2] \), \( [-2;0] \)
\( [0;-2] \), \( [0;2] \)
1003024105
Level:
C
Find the value of the parameter \( q\in\mathbb{R} \) for which the line \( x+2y-1=0 \) is a tangent of the hyperbola \( x^2-2y^2=q \).
\( -1 \)
\( 1 \)
\( 2 \)
\( -2 \)
\( \frac12 \)
1003024106
Level:
C
Find the value of the parameter \( q\in\mathbb{R} \) for which the line \( x+2y-1=0 \) is a tangent of the ellipse \( x^2+4y^2=q \).
\( \frac12 \)
\( \frac14 \)
\( 2 \)
\( 4 \)
\( 1 \)
1003024107
Level:
C
Find the value of the parameter \( p\in\mathbb{R} \) for which the line \( x+2y-1=0 \) is a tangent of the parabola \( y^2=2px \).
\( -\frac12 \)
\( \frac12 \)
\( 2 \)
\( -2 \)
\( 1 \)
1003024108
Level:
C
Find the value of the parameter \( q\in\mathbb{R} \) for which the line \( x+2y-1=0 \) is a tangent of the circle \( x^2+y^2=q^2 \).
\( \frac{\sqrt5}5 \)
\( 5 \)
\( \sqrt5 \)
\( \frac15 \)
\( 1 \)
1003024109
Level:
C
Find the set of values of the parameter \( c\in\mathbb{R} \) for which the line \( 3x+5y-c=0 \) is a secant of the ellipse \( 16x^2+25y^2=400 \).
\( (-25;25) \)
\( (-\infty;-25)\cup(25;\infty) \)
\( \{25\} \)
\( \{-25\} \)
\( (25;\infty) \)
1003024110
Level:
C
Find the value of the parameter \( q\in\mathbb{R} \) for which the line \( y=x+q \) is a tangent of the parabola \( y^2=6x \).
\( \frac32 \)
\( \frac23 \)
\( 0 \)
\( -\frac32 \)
\( -\frac23 \)
1003024111
Level:
C
The equation of a parabola is given by \( y^2 -12x -6y+57=0 \). Find the equation of a line, which passes through the vertex of this parabola and is parallel to the line \( 5x-3y-2=0 \).
\( 5x-3y-11 = 0 \)
\( -5x+3y-11 = 0 \)
\( 5x-3y-3 = 0 \)
\( 5x-3y+11 = 0 \)
\(-5x+3y-3 = 0 \)