IDZ Ryabushko 4.1 Option 9

No. 1. Drawing up canonical equations for the ellipse, hyperbola and parabola. a) For an ellipse with focus F and semi-major axis a, semi-minor axis b and eccentricity ε, the canonical equation has the form: ((x - x_F)²/a²) + ((y - y_F)²/b²) = 1 where (x_F, y_F) - coordinates of focus F.

b) For a hyperbola with foci F₁ and F₂, semi-major axis a, semi-minor axis b and eccentricity ε, the canonical equation has the form: ((x - x_F₁)²/a²) - ((y - y_F₁)²/b²) = 1 or - ((x - x_F₂)²/a²) + ((y - y_F₂)²/b²) = 1 where (x_F₁, y_F₁) and (x_F₂, y_F₂) are the coordinates of the foci F₁ and F₂.

c) For a parabola with focus F and parameter p, the canonical equation has the form: y² = 4px or x² = 4py depending on the orientation of the parabola relative to the coordinate axes.

Given: a) A(0;√3); B(√14/3;1); semimajor axis a = √14/3; semi-minor axis b = √11/3; eccentricity ε = √3/√14; the focus of F is on the y-axis and has coordinates (0,√8/3). Then the canonical equation of the ellipse will have the form: ((x - 0)²/(√14/3)²) + ((y - √8/3)²/(√11/3)²) = 1

b) The foci of the hyperbola 4x² - 5y² = 80 are on the x-axis and have coordinates (±√20.0); center of the circle A(0,-4). Consider a hyperbola with foci F₁(√20.0) and F₂(-√20.0), semi-major axis a and semi-minor axis b. From the equation of the hyperbola 4x² - 5y² = 80 it follows that a² = 20/4 = 5, b² = 80/5 = 16. The center of the hyperbola is at the point (0,0). This means that the canonical equation of the hyperbola has the form: ((x - 0)²/5²) - ((y - 0)²/4²) = 1 To find the equation of the desired circle, you need to find the point of intersection of the hyperbola and the line passing through the focus F₁ and point A. The coordinates of this point are (12/5, -12/5). The radius of the circle is equal to the distance from the center to this point, that is, √((12/5)² + (-12/5 + 4)²) = 4/5√13. Then the equation of the circle has the form: (x - 0)² + (y + 4)² = (4/5√13)²

c) Find the equation of a straight line that satisfies the condition. The distance from point M to straight line y = 7 is equal to 5 times the distance from point M to point A(4,-3). Let us denote the distance from point M to the line as d, and the distance from point M to point A as d₁. Then the condition can be written as follows: d = 5d₁ Using the formula for the distance from a point to a line, we obtain: |y - 7| = 5√((x - 4)² + (-3 - y)²) We divide it into two cases:

1. y - 7 = 5√((x - 4)² + (-3 - y)²)
2. y - 7 = -5√((x - 4)² + (-3 - y)²)

No. 4. Construction of a curve specified in the polar coordinate system: ρ = 4·sin 3φ. Note that the curve is symmetrical about the x-axis, so it is enough to draw it only for φ∈[0,π/2]. We start by plotting the function y = 4·sin 3x on the interval [0,π/2]. To do this, you can plot the function y = sin x, then multiply it by 4 and compress it along the x axis by 3 times. Then, for each value of x, we find the distance from the origin to the point on the graph with coordinates (x, 4·sin 3x), which is equal to ρ. We draw points with the obtained coordinates and connect them with a smooth curve.

No. 5. Construction of a curve specified parametrically: x = sin t, y = cos t (0 ≤ t ≤ 2π). To construct a curve, you can set the values ​​of the parameter t in the interval [0,2π] with a certain step, for example, t = 0, π/8, π/4, 3π/8, ..., 2π. For each value of t, we find the corresponding x and y, and mark a point on the plane with coordinates (x,y). Then we connect all the points with a smooth curve. The resulting curve is called a circle of unit radius with its center at the origin.

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IDZ Ryabushko 4.1 Option 9 is a set of problems in mathematics, which includes solving the following problems:

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

2. Find the equation of a circle passing through the given points and having a given center.

3. Find the equation of a line, each point of which is at a given distance from a given point and at a distance 5 times greater from the given line.

4. Construct a curve given in polar coordinates by the equation ρ = 4·sin 3φ.

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

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