Conics

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 \(y = 1\).

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.
\(k\in \left (-\infty ;-\frac{2} {3}\right )\cup \left (\frac{2} {3};\infty \right )\)
\(k\in \left [ -\frac{2} {3}; \frac{2} {3}\right ] \)
\(k\in \left (-\frac{2} {3}; \frac{2} {3}\right )\)
\(k\in \left (-\infty ;-\frac{2} {3}\right ] \cup \left [ \frac{2} {3};\infty \right )\)

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 \(p\colon 4x - 5y + 17 = 0.\)
\(q\colon 20x - 25y + 54 = 0\)
\(q\colon 20x - 25y - 27 = 0\)
\(q\colon 4x - 5y + 27 = 0\)
\(q\colon 4x -5y - 17 = 0\)

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.
\(p\colon \frac{x} {5} + \frac{y} {5} = 1\)
\(p\colon y = 5x\)
\(p\colon 2x + y = 5\)
\(\begin{aligned}[t] p\colon x& = 1 & \\y & = -1 + t\text{; }t\in \mathbb{R} \\ \end{aligned}\)

9000123108

Level: 
C
Find all the tangents to the hyperbola \(x^{2} - 2y^{2} = 8\) such that the angle between each tangent and the \(x\)-axis is \(45^{\circ }\).
\(y = x + 2\text{, }y = x - 2\text{, }y = -x + 2\text{, }y = -x - 2\)
\(y = x + 2\text{, }y = x - 2\)
\(y = x + 2\text{, }y = -x + 2\)
\(y = x + 2\)