Conics

1003024109

Level:

Find the set of values of the parameter

c \in R

for which the line

3 x + 5 y - c = 0

is a secant of the ellipse

16 x^{2} + 25 y^{2} = 400

(- 25; 25)

(- \infty; - 25) \cup (25; \infty)

{25}

{- 25}

(25; \infty)

1003024108

Level:

Find the value of the parameter

q \in R

for which the line

x + 2 y - 1 = 0

is a tangent of the circle

x^{2} + y^{2} = q^{2}

\frac{\sqrt{5}}{5}

5

\sqrt{5}

\frac{1}{5}

1

1003024107

Level:

Find the value of the parameter

p \in R

for which the line

x + 2 y - 1 = 0

is a tangent of the parabola

y^{2} = 2 p x

- \frac{1}{2}

\frac{1}{2}

2

- 2

1

1003024106

Level:

Find the value of the parameter

q \in R

for which the line

x + 2 y - 1 = 0

is a tangent of the ellipse

x^{2} + 4 y^{2} = q

\frac{1}{2}

\frac{1}{4}

2

4

1

1003024105

Level:

Find the value of the parameter

q \in R

for which the line

x + 2 y - 1 = 0

is a tangent of the hyperbola

x^{2} - 2 y^{2} = q

- 1

1

2

- 2

\frac{1}{2}

1003024104

Level:

Find the points of intersection of the circle

x^{2} + y^{2} = 4

and the line

x + y - 2 = 0

[0; 2]

[2; 0]

[0; - 2]

[- 2; 0]

[0; - 2]

[2; 0]

[0; 2]

[- 2; 0]

[0; - 2]

[0; 2]

1003024103

Level:

Find the equation of the circle, which passes through the points

A = [- 3; 2]

B = [- 1; 4]

and

C = [3; 0]

x^{2} + (y - 1)^{2} = 10

(x - 1)^{2} + y^{2} = 10

x^{2} + (y - 1)^{2} = \sqrt{10}

x^{2} + y^{2} = 10

x^{2} + (y + 1)^{2} = 10

1003024102

Level:

The equation of a parabola is given by

3 y^{2} + x - 12 y + 14 = 0

. Find the equation of its directrix.

x = - \frac{23}{12}

x = \frac{23}{12}

y = - \frac{23}{12}

y = \frac{23}{12}

x = - \frac{11}{6}

1003024101

Level:

The equation of a hyperbola with the center

S = [- 1; 3]

, the focus

F = [4; 3]

, and the vertex

A = [2; 3]

is given by:

\frac{(x + 1)^{2}}{9} - \frac{(y - 3)^{2}}{16} = 1

\frac{(x - 1)^{2}}{9} - \frac{(y + 3)^{2}}{16} = 1

\frac{(x + 1)^{2}}{9} + \frac{(y - 3)^{2}}{16} = 1

\frac{(x - 1)^{2}}{16} - \frac{(y + 3)^{2}}{9} = 1

\frac{(y - 3)^{2}}{16} - \frac{(x + 1)^{2}}{9} = 1

1003024203

Level:

Find the tangent equations from the point

L = [1; - 1]

to the parabola given by

y^{2} - 4 x + 2 y + 9 = 0

x - y - 2 = 0

x + y = 0

x + y - 2 = 0

x + y = 0

x + y + 2 = 0

x - y = 0

x - y + 2 = 0

- x + y = 0

- x + y - 2 = 0

x - y = 0

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