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Questions tagged [analytic-geometry]

Questions on the use of algebraic techniques for proving geometric facts. Analytic Geometry is a branch of algebra that is used to model geometric objects - points, (straight) lines, and circles being the most basic of these. It is concerned with defining and representing geometrical shapes in a numerical way.

Analytic geometry, also called coordinate geometry, mathematical subject in which algebraic symbolism and methods are used to represent and solve problems in geometry.

The importance of analytic geometry is that it establishes a correspondence between geometric curves and algebraic equations. This correspondence makes it possible to reformulate problems in geometry as equivalent problems in algebra, and vice versa; the methods of either subject can then be used to solve problems in the other.

Analytic geometry was introduced by René Descartes in $1637$ and was of fundamental importance in the development of the calculus by Sir Isaac Newton and G. W. Leibniz in the late $17^{th}$ cent. More recently it has served as the basis for the modern development and exploitation of algebraic geometry.

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Find the images of (1,0) under reflection in L?

Consider the line $$L = \{(x,y): x - 2y = 2\}$$ Find the images of $(1,0)$ under reflection in $L$? Thanks in advance.
Joe
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Vectors problem

can anyone help me with this problem: Is it possible to construct three vectors (a,b,c) in 3D, such that angle between a and b is 30 degrees, between a and c is 150 degrees, and between b and c is 30 degrees? If not, prove it.
Meee
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Formula to show square root between 2 values

Bear with me - it's been a while since I did this at school! I need to plot a curve in the form of a square root (kind of an 'r' shape if you will) I have 6 intervals along my x axis, and my maximum y value can be 180. Given a value of x, how can I…
Jeepstone
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Formula to plot a non-linear graph

Firstly, thank you very much in advance. I need to express a non-linear graph comprised (piecewise) of the following linear elements: A line from $(x=0,y=100)$ to $(x=10,y=100.5)$ A line from $(x=10,y=100.5)$ to $(x=30,y=99.5)$ A line from…
James Peterson
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Find vector and parametric equations of the line passing through the given point and perpendicular to the given plane.

Analytical Geometry of Space – Equation of Line Question: Find vector equation, parametric equations of the line passing through the point $(5,1,0)$ and perpendicular to the plane $2x-y+z=1$. Also find two other points on the line.
Honest reviews
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If in a general 2nd degree equation, there exists a xy term, is it always a hyperbola with rotated axes? Can it be ellipse or parabola?

For all the graphs I have plotted with non zero x², y² and xy term, I always have seen a hyperbola with a rotated axis. But under any condition can it be an ellipse? If yes, how?
Sarban Bhattacharya
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How to find the two opposite end coordinates of the perpendicular diameter?

If $x^2 + y^2 - x - 2y - 5 = 0$ is a circle equation and $(-1,-1),(2,3)$ are two opposite end of a diameter of this circle. Find the two opposite end coordinates of the perpendicular diameter.
Mohamed Asem
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Given points $A(7,2)$, $B(-4,2)$, $P(x,2x)$ such that the distance from $A$ to $P$ is equal to the distance from $B$ to $P$, find $x$

I have $A(7,2)$ and $B(-4,2)$, these 2 points in my graph. I have another point $P(x,2x)$. The distance between points $A$ and $P$ is equal to the distance between points $B$ and $P$. Find $x$.
Taki Tahmid
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Calculating angle of line and vector

What angle does the line pass through the points $A[1,1]$ and $B[−3, −3]$ with the vector $v=(−3,\sqrt{3})$? I proceed as follows: I determine the vector $AB = u = (-4, -4)$, then using the formula $$\cos (x) = \frac{u\cdot v}{\|u\|\cdot\|v\|}$$ I…
Aaron7
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Why do geometrical scaling laws work differently for empty vs. filled regions?

My question has to do with scaling laws, i.e., how surface areas scale as the square of the linear dimension, and volumes scale as the cube of the linear dimension. And, therefore, volumes should scale as the 3/2 power of areas. Examples: (1)…
Matt
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given point (2,6) and a line passes through point (3,0)

The question is: does the distance between the point $(2,6)$ to the that line could be $5$? is there a solution to the problem without computing? i would glad to know. thanks.
bori12
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proof: $l_i: a_i\times (x-x_i)=0$ (i=1, 2) intersects at exactly one point

given: $a_i, x_i\in R^3$, $a_1\times a_2 \neq 0$ and $det(a_1, a_2, x_2-x_1)$=0 prove: $l_i: a_i\times (x-x_i)=0$ (i=1, 2) intersects at exactly one point
CHEN WONG
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The equation of a plane orthogonal to another plane, when the intersection of the planes is a line from the xOy plane

Determine the equation of a plane orthogonal to the plane: $ (P) : 3x-y+3z-2=0$ and whose intersection with $(P)$ is a line from the $xOy$ plane. My idea is the following: since the planes are orthogonal it means that the dot product of the normal…
Andrei0408
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A series of equations

ax+y-2=0 x-ay+3=0 2x+y-a=0 Find a if (a,a) lies inside the triangle formed by these three lines
The Polymath
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Find the equation from perpendicular and line equation

Find the equation of the line which is at right angles to $3x + 4y =12$, such that its perpendicular distance from the origin is equal to the length of the perpendicular from $(3, 2)$ on the given line.​ How to solve this?
Apurba Adhikari
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