Conics

2010005906

Level:

Find all the tangents to the hyperbola $y^{2} - 4x^{2} = 12$ such that the angle between each tangent and the $x$-axis is $45^{\circ }$.

$y = x + 3,\ y = x - 3,\ y = -x + 3,\ y = -x - 3$

$y = x + 3,\ y =- x - 3$

$y = x + 3,\ y = x - 3$

$y = x + 3$

2010005905

Level:

Find the tangent line $r$ to the parabola $6(x+1) = (y-3)^{2}$, so that the tangent $r$ is parallel to the line $q\colon 3x - 2y + 7 = 0.$

$r\colon 3x - 2y + 11 = 0$

$r\colon 3x - 2y - 7 = 0$

$r\colon 3x - 2y - 13 = 0$

$r\colon 3x -2y + 13 = 0$

2010005904

Level:

Find a true statement about the ellipse \[ 9 x^{2} + y^{2} + 18x = 0. \]

The tangent to the ellipse can pass through any point on the line $x = 1$.

The tangent to the ellipse can pass through any point on the line $y = 1$.

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 $y = -1$.

2010005903

Level:

Find the values of the real parameter $r$ which ensure that the line $x = r$ is a tangent to the circle \[ x^{2} + y^{2} - 4x + 10y + 4 = 0. \]

$\{-3;7\}$

$\{ - 7;3\}$

$\{ - 10;0\}$

$\{ 0;10\}$

Find the distance between the points of intersection of the given hyperbola with the given straight line $q$. \[ H\colon \frac{\left (y+6\right )^{2}} {10} -\frac{\left (x-5\right )^{2}} {6} = 1;\quad q\colon y+1 = 0 \]

$6$

$8$

$10$

$12$

2010005901

Level:

Find the distance between the points where the $y$-axis intersects the following hyperbola. \[ H\colon \frac{\left (y+3\right )^{2}} {36} -\frac{\left (x+4\right )^{2}} {9} = 1 \]

$20$

$16$

$10$

$8$

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2010005906

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2010005903

2010005902

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Hyperbola and Its Equation I

Parabola and Its Equation I

Conic Sections and Their Equation

Ellipse and Its Equation I

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