1003024102
Level:
B
The equation of a parabola is given by \( 3y^2+x-12y+14=0 \). Find the equation of its directrix.
\( x=-\frac{23}{12} \)
\( x=\frac{23}{12} \)
\( y=-\frac{23}{12} \)
\( y=\frac{23}{12} \)
\( x=-\frac{11}{6} \)
Conics
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:
\( \frac{(x+1)^2}{9}-\frac{(y-3)^2}{16} =1 \)
\( \frac{(x-1)^2}{9}-\frac{(y+3)^2}{16} =1 \)
\( \frac{(x+1)^2}{9}+\frac{(y-3)^2}{16} =1 \)
\( \frac{(x-1)^2}{16}-\frac{(y+3)^2}{9} =1 \)
\( \frac{(y-3)^2}{16}-\frac{(x+1)^2}{9} =1 \)
1003024203
Level:
C
Find the tangent equations from the point \( L=[1;-1] \) to the parabola given by \( y^2-4x+2y+9=0 \).
\( x-y-2=0 \), \( x+y=0 \)
\( x+y-2=0 \), \( x+y=0 \)
\( x+y+2=0 \), \( x-y=0 \)
\( x-y+2=0 \), \( -x+y=0 \)
\( -x+y-2=0 \), \( x-y=0 \)
1003024202
Level:
C
Find the tangent equations from the point \( M=[0;0] \) to the ellipse given by \( x^2+2y^2-8x+4y+12=0\).
\( x-5y=0 \), \( x+y=0 \)
\( 5x-y=0 \), \( x+y=0 \)
\( x+5y=0 \), \( x-y=0 \)
\( -x-5y=0 \), \( -x+y=0 \)
\( 5x+y=0 \), \( x-y=0 \)
1003024201
Level:
C
Find the tangent equations from the point \( K=[5;0] \) to the circle given by \( (x-1)^2+(y-2)^2=4 \).
\( 4x+3y-20=0 \), \( y=0 \)
\( 3x+4y-20=0 \), \( y=0 \)
\( 4x+3y+20=0 \), \( y=0 \)
\( -4x+3y-20=0 \), \( y=0 \)
\( -3x+4y-20=0 \), \( y=0 \)
1003024008
Level:
B
Decide which of the given equations describes a hyperbola.
\( 4x^2-9y^2+18y-45=0 \)
\( 4x^2+9y^2-8x-36y+4=0 \)
\( x^2+y^2-2x+4y-4=0 \)
\( x^2+y^2-12x+40=0 \)
\( x^2-2x-4y+1=0 \)
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 \( \left[\frac32;\frac{-3}2\right] \)
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 \( [3;4] \)
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 \( \left[0;\frac14\right] \)
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