📘 analytic geometry

1. Problem: Find the coordinates of point $P(4,-1)$ after rotation about the origin by $180^\circ$. Formula: Rotation by $180^\circ$ transforms $(x,y)$ to $(-x,-y)$.

1. مسئله: نوع رویه داده شده به صورت $$x^2 + z^2 - 8x - (1 - 2)y + 2z + 13 = 0$$ را مشخص کنیم و شکل آن را رسم کنیم. 2. ابتدا معادله را بازنویسی می‌کنیم:

1. **Problem statement:** Determine the mutual position of the lines \(g\) and \(h\) and find the intersection point if they intersect. Given:

1. **Problem statement:** Given two lines AB and CD that are perpendicular and intersect at point (1, 4). Line AB crosses the x-axis at (3, 0). We need to find the coordinates of p

1. **Problem statement:** Calculate the coordinates of point I given points E and F of a cuboid and vectors \( \mathbf{u} = \begin{pmatrix}-1 \\ 2 \\ 1.5\end{pmatrix} \) and \( \ma

1. **Problem statement:** Given the family of lines $g_a:\ \vec{x} = \begin{pmatrix}5 \\ 1 \\ 4\end{pmatrix} + r \begin{pmatrix}a \\ 2 \\ 4 - 2a\end{pmatrix}, \ r,a \in \mathbb{R}$

1. **Problem Statement:** Determine if the statements about the two lines \(r\) and \(s\) in 3D space are true or false. Given:

1. **Statement:** Determine if the lines \(r\) and \(s\) are concurrent (intersect at a point). 2. **Given:**

1. **Problem statement:** Gegeben ist die Gerade $$g: \vec{x} = \begin{pmatrix} 1 \\ 1 \\ 0 \end{pmatrix} + s \cdot \begin{pmatrix} 5 \\ -4 \\ 2 \end{pmatrix}$$.

1. **State the problem:** Given points A(-3, 7) and B(5, -2), find the length of the line segment AB and the midpoint of AB. 2. **Length of the line segment formula:**

1. **Problem:** Find the equation of the parabola with vertex at (5, -2) and focus at (-5, -2). 2. **Recall the formula:** For a parabola with a horizontal axis, the vertex form is

1. **State the problem:** Find the equation of the parabola with vertex at $(5, -2)$ and focus at $(-5, -2)$. 2. **Recall the parabola definition:** A parabola is the set of points

1. **State the problem:** Find the equation of the tangent line to a circle at a given point on the circle in rectangular coordinates. 2. **Recall the circle equation:** A circle w

1. **State the problem:** Find the Cartesian equation of the line passing through the point $(-2,4,-5)$ and parallel to the line given by $$\frac{x+3}{3} = \frac{y-4}{5} = \frac{z+

1. **State the problem:** Find the point where the line perpendicular to \(L: \vec{r} = (1,-5) + s(3,5)\) passing through \(P(2,0)\) intersects the y-axis. 2. **Recall the line fro

1. Statement of the problem: We are given the equation $y^2 - 4x + h x = 0$ and we will analyze its conic type and key parameters.\n 2. Formula and important rules: The standard fo

1. **Problem statement.** Analyze the function $f(x)=\sqrt{8.58^2 - \frac{(x-11.1)^2}{1.24^2}}$ and describe domain, range, intercepts, and maximum; identify its relation to an ell

1. **بيان المسألة:** حدد العدد الحقيقي $m$ لكي تكون النقاط $A(1, m^2)$ و $B(1-5m, 1)$ و $C(5, 1)$ متزنة (أي تقع على نفس المستقيم).

1. **Problem statement:** We have a line $L$ from part 2 (assumed parametric form $x=1+t$, $y=2-t$, $z=3+2t$) and a plane $P$ with equation $x + 3y - z = -4$.

1. **Problem statement:** Given points $A_1(3,5,4)$, $A_2(5,8,3)$, $A_3(1,2,-2)$, and $A_4(-1,0,2)$, find: a) Equation of plane $A_1A_2A_3$.