IDZ Ryabushko 4.1 Option 3

No. 1. Below are the canonical equations for several types of curves, where A and B are points lying on the curve, F is the focus, a is the major (real) semi-axis, b is the minor (imaginary) semi-axis, ε is the eccentricity, y = ± k x - equations asymptote of the hyperbola, D - directrix of the curve, 2c - focal length.

a) Ellipse equation: $ \frac{(x-x_0)^2}{a^2} + \frac{(y-y_0)^2}{b^2} = 1 $, where $(x_0,y_0) $ - coordinates of the center of the ellipse.

b) Hyperbola equation: $ \frac{(x-x_0)^2}{a^2} - \frac{(y-y_0)^2}{b^2} = 1 $.

c) Parabola equation: $ y = a(x-x_0)^2 + y_0 $.

Given: a) $A(3;0)$, $B(2;\sqrt{5}/3)$; b) $k=3/4$, $\varepsilon=5/4$; c) $D: y=-2$.

No. 2. Below is the equation of a circle with center at point $A(x_0,y_0)$: $(x-x_0)^2 + (y-y_0)^2 = r^2$, where $r$ is the radius of the circle.

Given: Focuses of hyperbole $24y^2 - 25x^2 = $600; $A(-8;0)$.

No. 3. The equation of a line, each point of which is three times greater from the line $y=-2$ than from the point $A(5;0)$, has the form $y=mx+b$, where $m$ is the slope, $b$ is a free member.

No. 4. The curve defined in the polar coordinate system by the equation $\rho=2\sin(2\varphi)$ has the form of a cardioid.

No. 5. The curve given by the parametric equations $x(t)=\cos(t)$, $y(t)=\sin(2t)$ for $0\leq t\leq 2\pi$ is the Bernoulli lemniscate.

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"IDZ Ryabushko 4.1 Option 3" is a digital product containing tasks from various areas of mathematics, intended for students and schoolchildren studying mathematics at an advanced level. This product includes assignments on geometry, algebra, probability theory, and other topics.

In particular, in Ryabushko's IDZ 4.1 Option 3 there are tasks for composing canonical equations of an ellipse, hyperbola and parabola, using data on points, foci, semi-axes, eccentricity, asymptotes and other characteristics of curves. Also in the tasks there are equations of circles, lines and curves specified in polar and parametric coordinates.

The product has a beautiful HTML design and convenient navigation through tasks, which facilitates the learning process and increases the efficiency of learning the material. All the necessary information to complete the tasks is presented in a clear format with detailed explanations and conditions, which makes the learning process more interesting and effective.

The presence of "IDZ Ryabushko 4.1 Option 3" in the digital goods store allows everyone to receive a quality education in the field of mathematics.

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IDZ Ryabushko 4.1 Option 3 is a math task that consists of five different tasks.

1. It is necessary to compose canonical equations for the ellipse, hyperbola and parabola passing through points A and B, with given foci and parameters of the figures.

2. It is required to write down the equation of a circle that passes through given points and has a center at point A, for given foci of the hyperbola.

3. It is necessary to create an equation of a straight line, each point of which is three times greater from the straight line y = –2 than from the point A(5;0).

4. It is required to construct a curve specified in the polar coordinate system by the equation ρ = 2·sin 2φ.

5. It is necessary to construct a curve defined by parametric equations, where t varies from 0 to 2π.

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