IDZ Ryabushko 4.1 Option 8

No. 1. The canonical equation of the ellipse has the form: $$\frac{(x-x_0)^2}{a^2}+\frac{(y-y_0)^2}{b^2}=1,$$ where $(x_0 ,y_0)$ are the coordinates of the center of the ellipse, $a$ and $b$ are the lengths of the major and minor semi-axes, respectively. The canonical equation of a hyperbola has the form: $$\frac{(x-x_0)^2}{a^2}-\frac{(y-y_0)^2}{b^2}=1,$$ where $(x_0 ,y_0)$ are the coordinates of the center of the hyperbola, $a$ and $b$ are the lengths of the major and minor semi-axes, respectively. The canonical equation of a parabola has the form: $$y = a(x-x_0)^2+y_0,$$ where $(x_0,y_0)$ are the coordinates of the vertex of the parabola, $a$ is a parameter that determines the direction and shape of the parabola.

For given points $A$ and $B$, focus $F$ and eccentricity $\varepsilon$, the canonical equation of the ellipse has the form: $$\frac{(x-x_F)^2}{a^2}+\frac{( y-y_F)^2}{b^2}=1,$$ where $a$ and $b$ are determined from the relations $a = \frac{BF}{2}$, $b = \sqrt{a^2 - c^2}$, where $c = FB$, and the focus coordinates $F$ are calculated by the formula $F(x_F, y_F) = \left(x_A + \frac{\varepsilon a}{\sqrt{1-\ varepsilon^2}}, y_A\right)$.

For given parameters, the canonical hyperbola equation has the form: $$\frac{(x-x_F)^2}{a^2}-\frac{(y-y_F)^2}{b^2}=1,$$ where $a$ and $b$ are determined from the relations $a = \frac{1}{2\varepsilon}$, $b = \sqrt{a^2 + c^2}$, where $c = \sqrt{a^ 2 + b^2}$, and the focus coordinates $F$ are calculated by the formula $F(x_F, y_F) = \left(x_A + \frac{\varepsilon a}{\sqrt{1+\varepsilon^2}}, y_A\right)$.

For a given directrix $D$, the canonical equation of a parabola has the form: $$4p(y-y_0) = (x-x_0)^2,$$ where $p$ is the distance from the vertex of the parabola to the directrix, and the coordinates of the vertex of the parabola are $(x_0 ,y_0)$ are calculated as the midpoint of the segment connecting point $A$ and the point of intersection of the directrix with the axis $Oy$.

No. 2. The equation of a circle with center at point $A(x_A,y_A)$ and radius $r$ has the form: $$(x-x_A)^2+(y-y_A)^2=r^2.$$ The center of the circle lies on axis $Oy$, so its coordinate is $y_A=-2$. The radius of the circle can be found by replacing $x$ and $y$ in the hyperbola equation with the coordinates of its vertices, we get $r = \sqrt{(0-x_A)^2+(-2-y_A)^2}$. Thus, the equation of a circle has the form: $(x-0)^2+(y+2)^2=\left(\sqrt{(0-x_A)^2+(-2-y_A)^2}\right )^2.$ Substituting $x_A=0$ and $y_A=-2$, we obtain the final equation of the circle: $$x^2+(y+2)^2=68.$$

No. 3. Let the point $M(x,y)$ be located at a distance of $3d$ from the straight line $x=-5$, where $d$ is the distance from the point $M$ to the point $A(6,1)$. Then the distance from point $M$ to point $A$ is $\frac{d}{3}$, and we can write the equation of a circle with center at point $A$ and radius $\frac{d}{3}$: $$(x-6)^2 + (y-1)^2 = \left(\frac{d}{3}\right)^2.$$ Also, point $M$ lies on the perpendicular dropped from point $ A$ to the straight line $x=-5$. The equation of this perpendicular is $x=6$, so the $x$ coordinate of point $M$ is $6$. Thus, the equation of the line passing through the point $M$ is: $$x=6, \quad (y-1)^2 = \left(\frac{d}{3}\right)^2 - (x -6)^2.$$

No. 4. Let's move from polar coordinates $(\rho, \varphi)$ to Cartesian coordinates $(x,y)$ using the formulas $x=\rho\cos\varphi$ and $y=\rho\sin\varphi$. Substituting $\rho=3(1-\cos^2\varphi)$, we obtain the equation of the curve in Cartesian coordinates: $$x^2 + y^2 = 9(1-\cos^2\varphi)^2.$ $ This equation describes a curve called a "cardioid".

No. 5. For given parametric equations $x=f(t)$, $y=g(t)$, the curve can be found by plotting the dependence of $y$ on $x$ as the parameter $t$ changes from $0$ to $2\pi $.

For example, consider a parametrically defined curve: $$x = \cos t, \quad y = \sin t.$$ For $t=0$ the curve is at point $(1,0)$, for $t=\frac {\pi}{2}$ - at point $(0,1)$, for $t=\pi$ - at point $(-1,0)$, and so on. The graph of this curve is a circle of unit radius centered at the origin.

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"IDZ Ryabushko 4.1 Option 8" is a digital file in pdf format containing tasks on various topics in mathematics, including algebra. The file is intended for use by students when completing individual homework. The assignments may include examples of solving equations and systems of equations, finding derivatives, integrals, constructing graphs of functions, and other topics from the field of mathematics.

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IDZ Ryabushko 4.1 Option 8 is a math task that includes the following tasks:

1. Compose canonical equations for an ellipse, hyperbola and parabola passing through given points and having given parameters (major and minor semi-axes, eccentricity, focal length, etc.).

2. Find the equation of a circle passing through the vertex of a hyperbola and having a center at a given point.

3. Write an equation for a straight line that is at a distance three times greater from a given straight line than from a given point.

4. Construct a curve in polar coordinates given by the equation ρ = 3·(1 - cos^2φ).

5. Construct a curve given by parametric equations (0 ≤ t ≤ 2π).

This task is designed to test knowledge and skills in the field of analytical geometry and mathematical analysis.

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