Geometric properties of second-order curves and their canonical equations

$x^{2} + y^{2} + 4y = 0$

$x^{2} + y^{2} + 5x - 5y + 12 = 0$

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

$7x^{2} + 7y^{2} - 2x - 7y - 1 = 0$.

to have no common points with the circle $(x - a)^{2} + (y - b)^{2} = R^{2}$

to have two common points with this circle

to be tangent to this circle.

Find the equation of the tangent line drawn to the circle $(x - 1)^{2} + (y + 2)^{2} = 25$ at the point $M(- 3, 1)$.

Find the equations of the tangent lines to the circle $(x - 1)^{2} + (y + 1)^{2} = 9$ passing through the point $M(1, 4)$.

$(x - 1)^{2} + (y - 2)^{2} = 18, (x - 5)^{2} + (y - 6)^{2} = 2$

$(x + 1)^{2} + (y - 1)^{2} = 45, (x - 1)^{2} + (y - 5)^{2} = 5$.

$x^{2} + (y + 2)^{2} \leqslant 4$

$\left(x + \frac{1}{2}\right)^{2} + \left(y - \frac{3}{2}\right)^{2} > 25$

$x^{2} + y^{2} + 3x < 0, y < 0$

$- 1 \leqslant x^{2} + y^{2} - 2x + 2y \leqslant 7$

$\frac{x^{2}}{16} + \frac{y^{2}}{9} \leqslant 1$

$\frac{x^{2}}{4} + \frac{y^{2}}{9} > 1$

$1\leqslant \frac{x^{2}}{9} + y^{2} \leqslant 9$

$4x^{2} - 4x + 9y^{2} + 6y + 1 < 0$

$\sqrt{(x - 1)^{2} + y^{2}} + \sqrt{(x + 1)^{2} + y^{2}} < 6$

$\sqrt{x^{2} + (y - 1)^{2}} + \sqrt{x^{2} + (y + 1)^{2}} > 4$

$\frac{x^{2}}{16} - \frac{y^{2}}{9} \leqslant 1$

$\frac{x^{2}}{4} - \frac{y^{2}}{9} \geqslant 1$

$\sqrt{x} + \sqrt{y} \leqslant 2$

$\left|\frac{x^{2}}{4} - \frac{y^{2}}{36}\right| < 1$

$\left|3x^{2} - 9y^{2}\right| > 1$

$\sqrt{(x - 2)^{2} + y^{2}} - \sqrt{(x + 2)^{2} + y^{2}} < 2$

$y^{2} \leqslant 4x$

$y^{2} > 6x$

$x \leqslant y^{2} \leqslant 3x$

$- 2x - x^{2} < y^{2} < - 2x$.

$x = 3\cos t, y = 3\sin t, 0 \leqslant t < 2\pi$

$x = 1 + 2\cos t, y = 2 + 2\sin t, 0 \leqslant t < 2\pi$

$x = \cos t, y = \sin t, 0 \leqslant t < \pi$?

$r = 1$

$r = \frac{1}{1 - 2\cos \varphi }$

$r = \frac{3}{2 - \cos \varphi }$

$r = \frac{1}{\sin^{2}(\varphi / 2)}$.

$\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1, a > b > 0$

$\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1, b > a > 0$

$9x^{2} + 25y^{2} = 225$

$4x^{2} + y^{2} = 1$

$A(1, 1 / 6)$

$A(1 / 13, 1 / 13)$

$A(1 / 6, - 1 / 24)$

the distance between the vertices lying on the major axis equals 16, and the distance between the foci equals 10

the chord joining two vertices of the ellipse has length 5 and is inclined to its major axis at an angle $\arcsin (3 / 5)$

the foci of the ellipse are the points $(\pm 1, 0)$, and the point $(\sqrt{3}, \sqrt{3} / 2)$ belongs to the ellipse

the foci of the ellipse are the points $(\pm 2, 0)$, and the directrices are the lines $x = \pm 18$

the distance from a directrix to the nearest vertex equals 4, and to the vertex lying on the axis $Oy$ equals 8

the triangle with vertices at the foci and at the end of the minor axis is equilateral, and the diameter of the circle passing through the center and two vertices of the ellipse equals 7

the segment of the axis $Ox$ between the focus $F_{1}$ and the far vertex $A$ of the major axis is bisected by the second focus $F_{2}$, and the distance from $F_{2}$ to the line passing through $A$ and the vertex of the minor axis equals $1 / \sqrt{17}$

the directrices of the ellipse are the lines $x = \pm 4$, and the quadrilateral with vertices at the foci and the ends of the minor axis is a square

the eccentricity of the ellipse equals $\sqrt{7} / 4$, and the quadrilateral whose vertices are the vertices of the ellipse is circumscribed about a circle of radius 4.8.

the distance between the foci equals the arithmetic mean of the lengths of the axes

the segment between a focus and the far vertex of the major axis is divided by the second focus in the ratio $2: 1$

the distance from a focus to the far vertex of the major axis is 1.5 times the distance to the vertex of the minor axis

the segment between the foci is seen from the end of the minor axis at a right angle

the major axis is seen from the end of the minor axis at an angle of $120^{\circ }$

the segment between a focus and the far vertex of the major axis is seen from the end of the minor axis at a right angle

the sides of a square inscribed in the ellipse pass through the foci of the ellipse.

at a right angle

at an angle of $60^{\circ }$

at the largest angle.

with common foci $(\pm c, 0)$

with common directrices $x = \pm d$ and common center at the origin.

the points $F_{1}(5, 1)$ and $F_{2}(- 1, 1)$ are the foci, and the line $x = 31 / 3$ is one of the directrices

the point $F(- 6, 2)$ is one of the foci, the point $A(2, 2)$ is an endpoint of the major axis, and the eccentricity equals $2 / 3$

the axes of the ellipse are parallel to the coordinate axes, the points $A(4, 0)$ and $B(0, 4)$ belong to the ellipse, and the point $B$ is at distance $3\sqrt{2}$ from one of the foci and at distance 6 from the corresponding directrix.

Prove that the quantity $\frac{1}{\left|OA\right|^{2}} + \frac{1}{\left|OB\right|^{2}}$ is constant for all possible pairs of points $A$ and $B$.

Find the largest and smallest values of the length of the segment $AB$.

$\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$

$\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = - 1$

$\frac{x^{2}}{16} - \frac{y^{2}}{9} = 1$

$y^{2} - x^{2} = 1$

$xy = 1$

$xy = - 2$.

$A(1 / 8, - 1 / 8)$

$A(1, 1)$

$A(1, 7)$

$A(- 1 / 2, 0)$.

the distance between the vertices equals 10, and the distance between the foci equals 12

the length of the real axis equals 1, and the point $(1, 3)$ belongs to the hyperbola

the directrices of the hyperbola are the lines $x = \pm \sqrt{5 / 6}$, and the point $(- 9, 4)$ belongs to the hyperbola

the length of the imaginary semi-axis equals 1, and a vertex of the hyperbola divides the segment between the foci in the ratio $4: 1$

the eccentricity of the hyperbola equals $7 / 5$, and the distance from a vertex to the nearest focus equals 2

the point $(7, - 2\sqrt{3})$, belonging to the hyperbola, is at distance $4\sqrt{7}$ from the left focus

the angle between the asymptotes containing a focus equals $60^{\circ }$, and the distance from a directrix to the nearest vertex equals $\frac{3}{2}(2 - \sqrt{3})$

the point $(- 5 / 4, 3 / 2)$ belongs to the hyperbola, and the asymptotes are the lines $y = \pm 2x$

the point $(- 1, 3)$ belongs to the hyperbola, and the asymptotes are the lines $y = \pm 2x$.

its semi-axes are equal (an equilateral hyperbola)

the angle between the asymptotes containing a focus equals $120^{\circ }$

the asymptotes of the hyperbola are the lines $y = \pm 3x$.

the distances from the point $M(5, - 4)$, belonging to the hyperbola, to the directrices are in the ratio $2: 1$

the sum of the distances from the point $N(- 5, - 4)$ to the asymptotes of the hyperbola equals 20 / 3.

at a right angle

at an angle of $60^{\circ }$

at the largest angle.

with common foci $(\pm c, 0)$

with common directrices $x = \pm d$ and common center at the origin

with common asymptotes $y = \pm kx$.

the points $F_{1}(3, - 2)$ and $F_{2}(5, - 2)$ are the foci, and the line $x = 7 / 2$ is one of the directrices

the point $F(1, 3)$ is one of the foci, the point $A(- 4, 3)$ is a vertex, and the eccentricity equals $3 / 2$

the point $F(0, 0)$ is one of the foci, and the lines $x\pm y + 2 = 0$ are the asymptotes.

the product of the distances from any point of the hyperbola to its asymptotes

the area of a parallelogram, one of whose vertices lies on the hyperbola and two of whose sides lie on the asymptotes.

$y^{2} = 2px, \quad p > 0$

$y^{2} = - px, \quad p > 0$

$y^{2} = 6x$

$y^{2} = - 3x$

$y = x^{2}$

$y = - \sqrt{3} x^{2}$.

$(5, - 7)$

$(8, 9)$

$(5 / 2, - 5)$?

the point $(5, - 5)$ belongs to the parabola

the distance from the focus to the directrix equals 12

the length of the chord passing through the focus at an angle of $45^{\circ }$ to the axis of the parabola equals 18.

the line passing through the point $M$ and the focus of the parabola makes an angle of $60^{\circ }$ with the axis $Ox$

the area of the triangle with vertices at the required point $M$, the focus of the parabola, and the point where the axis of the parabola meets the directrix equals 5

the distance from the point $M$ to the vertex of the parabola equals the distance from $M$ to the focus

the distances from the point $M$ to the vertex of the parabola and to the focus of the parabola are in the ratio $8: 7$.

with the positive direction of the axis $Ox$

with the negative direction of the axis $Ox$

with the positive direction of the axis $Oy$

with the negative direction of the axis $Oy$.

having common focus $(0, 0)$ and symmetric with respect to the axis $Ox$

having common directrix $x = 0$ and symmetric with respect to the axis $Ox$.

the point $F(7, 0)$ is the focus, and the line $x = 1$ is the directrix

the point $F(7, 0)$ is the focus, and the line $x = 8$ is the directrix

the point $F(0, 1)$ is the focus, the parabola is symmetric with respect to the axis $Oy$, and is tangent to the axis $Ox$

the axis of the parabola is parallel to the axis $Oy$, the focus lies on the axis $Ox$, the parabola passes through the origin and cuts off on the axis $Ox$ a segment of length 6.