IDZ Ryabushko 4.1 Option 22

No. 1 Create a canonical equation for: a) an ellipse: (x + p)²/a² + (y + q)²/b² = 1, where (p, q) are the coordinates of the center of the ellipse, a and b are the lengths of the major and minor axle shafts respectively. b) hyperbolas: (x + p)²/a² - (y + q)²/b² = 1, where (p, q) are the coordinates of the center of the hyperbola, a and b are the lengths of the major and minor semi-axes, respectively. c) parabolas: y² = 2px, where p is the parabola parameter that determines the distance from the focus to the vertex.

For point A(-6;0) and ε = 2/3: a) ellipse: (x + 6)²/81 + y²/36 = 1. b) hyperbola: (x + 6)²/9 - y²/ 16 = 1. c) The condition is not correct. The coordinates of the points are incorrect. Not resolved.

For point A(√8;0): a) ellipse: x²/2 + y²/((2/3)·2) = 1. b) hyperbola: x²/2 - y²/((2/3)·2 ) = 1. c) parabolas: y² = 8x.

For D: y = 1: a) ellipse: does not exist. b) hyperbolas: (x - 4)²/9 - y²/8 = 1. c) parabolas: y² = 8(x - 3).

No. 2 The parabola equation x² = -2(y + 1) has a vertex at point A(0, -1). The center of the circle coincides with the vertex of the parabola, therefore it is located at point A(0, -1). To find the radius of the circle, it is necessary to find the distance from point B(2, -5) to point A(0, -1), which is equal to √((2 - 0)² + (-5 + 1)²) = √20. Thus, the equation of the desired circle is: (x - 0)² + (y + 1)² = 20.

No. 3 Let the coordinates of point M on the desired line be equal to (x, y). Then the ratio of the distances from point M to points A(3,-2) and B(4,6) is equal to 3/5 and can be written as: (x - 3)² + (y + 2)² / ((x - 4 )² + (y - 6)²) = 9/25. Opening the brackets, bringing similar ones and transforming the equation, we obtain the desired equation of the straight line: 16x - 9y - 94 = 0.

No. 4 The curve is given in polar coordinates: ρ = 2·cos 4φ. Convert to Cartesian coordinates: x = ρ·cos φ, y = ρ·sin φ. Substituting expressions for ρ and simplifying, we obtain the equation of the curve in Cartesian coordinates: (x² + y²)² - 8x²y² = 16x².

No. 5 The required curve is specified parametrically by the equations: x = cos t, y = sin t. This equation parametrically defines a circle with a center at the origin and a radius of 1. To construct a curve, you can plot these parametric equations in Cartesian coordinates with t taking values ​​from 0 to 2π.

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IDZ Ryabushko 4.1 Option 22 is a digital product intended for secondary school students, containing tasks and exercises in mathematics. Tasks include:

No. 1. Drawing up canonical equations for the ellipse, hyperbola and parabola, as well as finding the basic parameters of the curve, such as points on the curve, focus, semi-axes, eccentricity, equations of asymptotes and directrixes.

No. 2. Finding the equation of a circle with its center at a given point A and passing through a given point B.

No. 3. Drawing up an equation of a straight line, each point of which satisfies the condition of the ratio of distances to given points.

No. 4. Constructing a curve specified in polar coordinates in Cartesian coordinates.

No. 5. Construction of a curve defined parametrically in the form of a circle with a center at the origin and radius 1.

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IDZ Ryabushko 4.1 Option 22 is a task consisting of five different problems in mathematics.

No. 1. This problem requires you to construct canonical equations for an ellipse, hyperbola, and parabola using given points, foci, semi-axes, and other parameters. For each curve, you also need to find the eccentricity, equations of asymptotes (for the hyperbola), directrix and focal length. The eccentricity values ​​and coordinates of the points for which equations need to be drawn up are given.

No. 2. In this problem, you need to write down the equation of a circle that passes through a given point B(2;-5) and has its center at the vertex of a parabola defined by the equation x^2 = -2(y+1).

No. 3. In this problem, you need to create an equation of a straight line, each point of which satisfies the condition: the ratio of the distances from point M to points A(3;-2) and B(4;6) is equal to 3/5.

No. 4. In this problem you need to plot a curve given in polar coordinates by the equation ρ = 2·cos 4φ.

No. 5. This problem requires you to graph a curve given by parametric equations where t ranges from 0 to 2π.

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