9000149703
Level:
A
Find the center of the following circle.
\[
x^{2} + y^{2} - 10x - 2y + 10 = 0
\]
\([5;1]\)
\([5;-1]\)
\([-5;1]\)
\([-5;-1]\)
Conics
9000149704
Level:
A
Find the center of the following ellipse.
\[
9x^{2} + 4y^{2} + 54x - 32y + 109 = 0
\]
\([-3;4]\)
\([-3;-4]\)
\([3;4]\)
\([3;-4]\)
9000149705
Level:
A
Find the center of the following ellipse.
\[
16x^{2} + 9y^{2} - 32x - 54y - 47 = 0
\]
\([1;3]\)
\([1;-3]\)
\([-1;3]\)
\([-1;-3]\)
9000149706
Level:
B
Find the center of the following hyperbola.
\[
4x^{2} - 3y^{2} + 8x - 30y - 49 = 0
\]
\([-1;-5]\)
\([-1;5]\)
\([1;-5]\)
\([1;5]\)
9000149707
Level:
B
Find the center of the following hyperbola.
\[
5x^{2} - 6y^{2} - 30x + 12y + 9 = 0
\]
\([3;1]\)
\([3;-1]\)
\([-3;1]\)
\([-3;-1]\)
9000149710
Level:
B
Find the vertex of the following parabola.
\[
x^{2} - 6x - 12y - 3 = 0
\]
\([3;-1]\)
\([3;1]\)
\([-3;1]\)
\([-3;-1]\)
9000123105
Level:
C
Find the all values of the real parameter
\(p\) which ensure
that the line \(q\colon y = x - 1\)
is a tangent to the parabola
\[
x^{2} = 2py.
\]
\(p = 2\)
\(p\in \{0;2\}\)
\(p = -2\)
\(p\in \{ - 2;0\}\)
9000123104
Level:
C
In the following list identify a line which is a tangent to the ellipse
\[
(x - 2)^{2} + \frac{y^{2}}
{9} = 1
\]
\(\begin{aligned}[t] p\colon x& = 3 + t, &
\\y & = 3;\ t\in \mathbb{R}
\\ \end{aligned}\)
\(p\colon x = 2\)
\(p\colon y = 3x\)
\(p\colon y = -x - 2\)
9000123107
Level:
C
In the following list identify a line such that the line has a unique intersection with
the hyperbola
\[
x^{2} - y^{2} = 5
\]
but the line is not the tangent to this hyperbola.
\(p\colon \frac{x}
{5} + \frac{y}
{5} = 1\)
\(p\colon y = 5x\)
\(p\colon 2x + y = 5\)
\(\begin{aligned}[t] p\colon x& = 1 &
\\y & = -1 + t\text{; }t\in \mathbb{R}
\\ \end{aligned}\)
9000123103
Level:
C
The ellipse
\[
5x^{2} + 9y^{2} = 45
\]
has tangent \(2x + 3y = 9\). Find the
values of the real parameter \(k\)
which ensure that the line \(y = kx + 3\)
is a secant for the ellipse.
\(k\in \left (-\infty ;-\frac{2}
{3}\right )\cup \left (\frac{2}
{3};\infty \right )\)
\(k\in \left [ -\frac{2}
{3}; \frac{2}
{3}\right ] \)
\(k\in \left (-\frac{2}
{3}; \frac{2}
{3}\right )\)
\(k\in \left (-\infty ;-\frac{2}
{3}\right ] \cup \left [ \frac{2}
{3};\infty \right )\)