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.
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How to check if a point is inside a rectangle?
There is a point $(x,y)$, and a rectangle $a(x_1,y_1),b(x_2,y_2),c(x_3,y_3),d(x_4,y_4)$, how can one check if the point inside the rectangle?
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Prove the theorem on analytic geometry in the picture.
I discovered this elegant theorem in my facebook feed. Does anyone have any idea how to prove?
Formulations of this theorem can be found in the answers and the comments. You are welcome to join in the ...
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How to straighten a parabola?
Consider the function $f(x)=a_0x^2$ for some $a_0\in \mathbb{R}^+$. Take $x_0\in\mathbb{R}^+$ so that the arc length $L$ between $(0,0)$ and $(x_0,f(x_0))$ is fixed. Given a different arbitrary $a_1$, ...
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What is the general equation of the ellipse that is not in the origin and rotated by an angle?
I have the equation not in the center, i.e.
$$\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1.$$
But what will be the equation once it is rotated?
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How to know if a point is inside a circle?
Having a circle with the centre $(x_c, y_c)$ with the radius $r$ how to know whether a point $(x_p, y_p)$ is inside the circle?
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Is there an equation to describe regular polygons?
For example, the square can be described with the equation $|x| + |y| = 1$. So is there a general equation that can describe a regular polygon (in the 2D Cartesian plane?), given the number of sides ...
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To which degree must I rotate a parabola for it to be no longer the graph of a function?
To which degree must I rotate a parabola for it to be no longer the graph of a function?
I have no problem with narrowing the question down by only concerning the standard parabola: $$f(x)=x^2.$$
I ...
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False proof: $\pi = 4$, but why?
Note: Over the course of this summer, I have taken both Geometry and Precalculus, and I am very excited to be taking Calculus 1 next year (Sophomore for me). In this question, I will use things that I ...
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What is the analogue of spherical coordinates in $n$-dimensions?
What's the analogue to spherical coordinates in $n$-dimensions? For example, for $n=2$ the analogue are polar coordinates $r,\theta$, which are related to the Cartesian coordinates $x_1,x_2$ by
$$x_1=...
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How to calculate the intersection of two planes?
How to calculate the intersection of two planes ?
These are the planes and the result is gonna be a line in $\Bbb R^3$:
$x + 2y + z - 1 = 0$
$2x + 3y - 2z + 2 = 0$
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Dog bone-shaped curve: $|x|^x=|y|^y$
EDITED: Some of the questions are ansered, some aren't.
EDITED: In order not to make this post too long, I posted another post which consists of more questions.
Let $f$ be (almost) the implicit curve$...
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Get the equation of a circle when given 3 points
Get the equation of a circle through the points $(1,1), (2,4), (5,3) $.
I can solve this by simply drawing it, but is there a way of solving it (easily) without having to draw?
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Why are we justified in using the real numbers to do geometry?
Context: I'm taking a course in geometry (we see affine, projective, inversive, etc, geometries) in which our basic structure is a vector space, usually $\mathbb{R}^2$. It is very convenient, and also ...
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Lampshade Geometry Problem
Today, I encountered a rather interesting problem in a waiting room:
$\qquad \qquad \qquad \qquad$
Notice how the light is being cast on the wall? There is a curve that defines the boundary between ...
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Derivation of the formula for the vertex of a parabola
I'm taking a course on Basic Conic Sections, and one of the ones we are discussing is of a parabola of the form
$$y = a x^2 + b x + c$$
My teacher gave me the formula:
$$x = -\frac{b}{2a}$$
as the ...
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