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:
\( [5;-3] \)
\( [-5;3] \)
\( [-5;-3] \)
\( [12;8] \)
\( [8;12] \)
Conics
1003020406
Level:
B
The equation of a hyperbola is given by \( \frac{x^2}4-\frac{y^2}3=1 \). Which of the following are the asymptote equations of the hyperbola?
\( y=\pm\frac{\sqrt3}2x \)
\( x=\pm\frac{\sqrt3}2y \)
\( y=\pm\frac34 x \)
\( y=\pm\frac43 x \)
\( y=\pm\frac{2\sqrt3}3 x \)
1003020405
Level:
B
A hyperbola has two asymptotes with the equations \( y=\frac13(x+2) \) and \(y=-\frac13(x+2)\). What are the coordinates of its centre?
\( [-2;0] \)
\( [2;0] \)
\( \left[-\frac13;0\right] \)
\( [0;2] \)
\( \left[\frac13;2\right] \)
1003020404
Level:
B
The equation of a hyperbola is given by \( 9x^2-16y^2-108x+96y+36=0\). The length of its semi-major axis is:
\( 4 \)
\( 16 \)
\( 3 \)
\( 9 \)
\( 25 \)
1003020403
Level:
A
An ellipse has the centre at \( S=[-1;3] \), the semi major axis with the length of \( 3 \) and the semi minor axis with the length of \( 2 \). The semi major axis is parallel to \( y \) axis. The equation of the ellipse is:
\( \frac{(x+1)^2}4+ \frac{(y-3)^2}9 = 1 \)
\( \frac{(x+1)^2}9 + \frac{(y-3)^2}4 = 1 \)
\( \frac{(x-1)^2}4+ \frac{(y+3)^2}9 = 1 \)
\( \frac{(x+1)^2}4-\frac{(y-3)^2}9 = 1 \)
\( \frac{(x+1)^2}4 + \frac{(y-3)^2}9 = -1 \)
1003020402
Level:
A
The equation of an ellipse is given by \( x^2+4y^2-6x+32y+48=0 \). The length of its semi-minor axis is:
\( \frac52 \)
\( 5 \)
\( 25 \)
\( \frac{25}4 \)
\( \frac{15}2 \)
1003020401
Level:
A
The equation of an ellipse is given by \( 5x^2+9y^2-30x-18y+9=0 \). The length of its semi-major axis is:
\( 3 \)
\( 5 \)
\( 9 \)
\( \sqrt5 \)
\( 14 \)
9000168710
Level:
B
Find the vertex of the hyperbola \(9x^{2} - 25y^{2} + 18x + 100y - 316 = 0\).
\([4;2]\)
\([2;2]\)
\([-1;-3]\)
\([-1;-1]\)
9000168709
Level:
B
Find the vertex of the hyperbola \(9x^{2} - 16y^{2} - 36x - 96y + 36 = 0\).
\([2;-6]\)
\([2;-7]\)
\([5;-3]\)
\([6;-3]\)
9000168702
Level:
A
Given the ellipse \(4x^{2} + 9y^{2} + 16x - 18y - 11 = 0\),
find the coordinates of the vertex on the minor axis.
\([-2;3]\)
\([-2;4]\)
\([0;1]\)
\([1;1]\)