9000149705
Level:
A
Find the center of the following ellipse.
\[
16x^{2} + 9y^{2} - 32x - 54y - 47 = 0
\]
\([1;3]\)
\([1;-3]\)
\([-1;3]\)
\([-1;-3]\)
Conic Sections
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.
\[
x^{2} - 6x - 12y - 3 = 0
\]
\([3;-1]\)
\([3;1]\)
\([-3;1]\)
\([-3;-1]\)
9000149709
Level:
B
Find the vertex of the following parabola.
\[
y^{2} - 12x + 4y + 64 = 0
\]
\([5;-2]\)
\([5;2]\)
\([-5;2]\)
\([-5;-2]\)
9000149708
Level:
B
Find the vertex of the following parabola.
\[
x^{2} + 8x - 4y + 24 = 0
\]
\([-4;2]\)
\([-4;-2]\)
\([4;2]\)
\([4;-2]\)
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.
\(k\in \left (-\infty ;-\frac{2}
{3}\right )\cup \left (\frac{2}
{3};\infty \right )\)
\(k\in \left [ -\frac{2}
{3}; \frac{2}
{3}\right ] \)
\(k\in \left (-\frac{2}
{3}; \frac{2}
{3}\right )\)
\(k\in \left (-\infty ;-\frac{2}
{3}\right ] \cup \left [ \frac{2}
{3};\infty \right )\)
9000123101
Level:
C
Find the values of the real parameter \(q\)
which ensure that the line \(y = q\)
is a tangent to the circle
\[
x^{2} + y^{2} + 4x - 8y + 4 = 0.
\]
\(\{0;8\}\)
\(\{ - 6;2\}\)
\(\{ - 8;0\}\)
\(\{ - 2;6\}\)
9000123102
Level:
C
Find a true statement about the ellipse
\[
x^{2} + 4y^{2} - 8y = 0.
\]
The tangent to the ellipse can pass through any point on the line
\(y = -1\).
The tangent to the ellipse can pass through any point on the line
\(x = 1\).
The tangent to the ellipse can pass through the point
\([-1;1]\).
The tangent to the ellipse can pass through any point on the line
\(y = 1\).
9000123106
Level:
C
Find the tangent line \(q\) to
the parabola \(4(y - 2) = (x + 1)^{2}\), so that the tangent \(q\)
is parallel to the line \(p\colon 4x - 5y + 17 = 0.\)
\(q\colon 20x - 25y + 54 = 0\)
\(q\colon 20x - 25y - 27 = 0\)
\(q\colon 4x - 5y + 27 = 0\)
\(q\colon 4x -5y - 17 = 0\)