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.
\( y=x^2+6x \)
\( y=x^2 \)
\( y=7x^2 \)
\( y=-x^2+4x \)
\( x=y^2 \)
Conics
1003024002
Level:
A
The equation of an ellipse is given by \( \frac{x^2}{48} +\frac{y^2}{36} = 1\). The point \( [-2;5] \) represents:
a point inside the ellipse
vertex of the ellipse
co-vertex of the ellipse
a point outside the ellipse
the centre of the ellipse
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?
\( [-2;2] \)
\( [2;1] \)
\( [0;12] \)
\( [3;0] \)
\( [\sqrt3;12] \)
1003020413
Level:
B
The equation of a parabola is given by \( x^2+6x+y+10=0 \). What are the coordinates of its vertex?
\( [-3;-1] \)
\( [3;1] \)
\( [3;-1] \)
\( [-1;-3] \)
\( [-1;3] \)
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?
\( [0;0] \)
\( [6;-2] \)
\( [3;0] \)
\( [-6;-2] \)
\( [0;-1] \)
1003020411
Level:
B
A parabola has its focus \( F=[5;0] \) and the directrix with the equation \( x=0 \). What are the coordinates of its vertex?
\( \left[\frac52;0\right] \)
\( [0;0] \)
\( \left[0;\frac52\right] \)
\( [-5;0] \)
\( \left[-\frac52;0\right] \)
1003020410
Level:
B
The equation of a parabola is given by \( x^2+4y-6x+3=0 \). What is the parameter value of this parabola?
\( 2 \)
\( 4 \)
\( -2 \)
\( -4 \)
\( \frac32 \)
1003020409
Level:
B
The equation of a parabola is given by \( y^2 = 4x \). What is the parameter value of this parabola?
\( 2 \)
\( 4 \)
\( \frac12 \)
\( \frac14 \)
\( 1 \)
1003020408
Level:
B
The equation of a hyperbola is given by \( \frac{x^2}9-\frac{y^2}{16}=1 \). What is the distance between its foci?
\( 10 \)
\( 5 \)
\( 16 \)
\( 9 \)
\( 4 \)
1003020407
Level:
B
The equation of a hyperbola is given by \( \frac{(x-5)^2}{12}-\frac{(y+3)^2}8=1 \). The centre of the hyperbola has the coordinates:
\( [5;-3] \)
\( [-5;3] \)
\( [-5;-3] \)
\( [12;8] \)
\( [8;12] \)