1003020412
Level:
B
A parabola passes through the points \( K=[0;0] \) and \( L=[6;-2] \) and it is symmetric with respect to \( y \)-axis. What are the coordinates of its vertex?
\( [0;0] \)
\( [6;-2] \)
\( [3;0] \)
\( [-6;-2] \)
\( [0;-1] \)
Conics
1003020413
Level:
B
The equation of a parabola is given by \( x^2+6x+y+10=0 \). What are the coordinates of its vertex?
\( [-3;-1] \)
\( [3;1] \)
\( [3;-1] \)
\( [-1;-3] \)
\( [-1;3] \)
1003024001
Level:
B
The equation of a hyperbola is given by \( 4x^2-y^2-12=0 \). Which of the following points lies on such hyperbola?
\( [-2;2] \)
\( [2;1] \)
\( [0;12] \)
\( [3;0] \)
\( [\sqrt3;12] \)
1003024003
Level:
B
Let \( A=[0;0] \), \( B=[1;7] \), \( C=[-1;-5] \) be points that all lie on a parabola. Find the equation of the parabola.
\( y=x^2+6x \)
\( y=x^2 \)
\( y=7x^2 \)
\( y=-x^2+4x \)
\( x=y^2 \)
1003024004
Level:
B
Decide whether the equation \( 4x^2+9y^2-8x+18y+13=0 \) (in \( x \)-\( y \) plane) describes:
a point
an ellipse
a parabola
a hyperbola
an empty set
1003024005
Level:
B
Decide whether the equation \( 4x^2+4y-8x=1\) (in \( x \)-\( y \) plane) describes:
a parabola
an ellipse
a circle
a hyperbola
point \( \left[0;\frac14\right] \)
1003024006
Level:
B
Decide whether the equation \( (x+3)^2+(y+4)^2=5 \) (in \( x \)-\( y \) plane) describes:
a circle with center \( [-3;-4] \)
a circle with center \( [3;4] \)
point \( [-3;-4] \)
a hyperbola
an ellipse with center \( [3;4] \)
1003024007
Level:
B
Decide whether the equation \( 2x^2-4y^2-6x-12y=0 \) (in \( x \)-\( y \) plane) describes:
a hyperbola
a circle
a parabola
an ellipse
point \( \left[\frac32;\frac{-3}2\right] \)
1003024008
Level:
B
Decide which of the given equations describes a hyperbola.
\( 4x^2-9y^2+18y-45=0 \)
\( 4x^2+9y^2-8x-36y+4=0 \)
\( x^2+y^2-2x+4y-4=0 \)
\( x^2+y^2-12x+40=0 \)
\( x^2-2x-4y+1=0 \)
1003024101
Level:
B
The equation of a hyperbola with the center \( S=[-1;3] \), the focus \( F=[4;3] \), and the vertex \( A=[2;3] \) is given by:
\( \frac{(x+1)^2}{9}-\frac{(y-3)^2}{16} =1 \)
\( \frac{(x-1)^2}{9}-\frac{(y+3)^2}{16} =1 \)
\( \frac{(x+1)^2}{9}+\frac{(y-3)^2}{16} =1 \)
\( \frac{(x-1)^2}{16}-\frac{(y+3)^2}{9} =1 \)
\( \frac{(y-3)^2}{16}-\frac{(x+1)^2}{9} =1 \)