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.
7,066 questions
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Maximizing the area of the triangle determined by foci of three parabolas, each touching all the three lines $x=0,y=0,x+y=2$
I am attempting to solve the question:
If three parabolas touch all the three lines: $x=0,y=0,x+y=2$ then what will be the maximum are of the triangle formed by joining their foci. The expected answer ...
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Difficulties with Set-Notation in Taxicab Geometry
The textbook I use introduced the Taxicab "circle" using the set-notation,
$$\{P\mid d_{T}(P,A)=1\}.$$
as the set of points, such that the distance between all the points and the center is ...
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Counting intersection points where multiple equations coincide at least twice (is there a known framework?)
I was plotting functions and this came to my mind:
Consider the three equations:
$$x=\text{a}\cos\theta$$
$$y=\text{a}\sin\theta$$
$$x^2+y^2=\text{a}^2$$
How do I obtain the general function below ...
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Why do the lines $y_1=x$ and $y_2=x+1$ never meet? [duplicate]
Somewhat stupid question, but why do the lines $y_1=x$ and $y_2=x+1$ never meet in a point?
Is it just because of algebra? Whatever $y_1$ is equal to, $y_2$ will be $y_1+1$?
Thanks.
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Find the range of $a$ such that there exists a line having no intersection with $2\cos x−\cos y=a$
Problem:
If there exists a line not parallel to the coordinate axes or $y=\pm x$ that does not intersect the curve
$$2\cos x - \cos y = a (a \ge 0)$$
Find the range of $a$.
This is a problem I came ...
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Finding the area of the shaded region inside the square and the rectangle in a more elegant way
The diagram shows a rectangle with side lenths $4$ and $8$ and a square with side length $5$. Three vertices of the square lie on three different sides of the rectangle, as shown. If the area of the ...
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If the vertices of triangle have rational coordinates, then prove that the triangle cannot be equilateral. [closed]
This Triangle is equilateral, and its coordinates are all rational:
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Modern rendering of traditional definition of straightness
A traditional definition (by Leibnizian mathematicians) had it that "A straight line segment is a line segment that is similar to all its parts". What would be a modern rendering of the ...
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A beautiful heart-shaped curve generated from an ellipse and a circle
As shown in the figure, the blue curve is an ellipse, and the red circle has the major axis of the ellipse as its diameter. $F$ is a focus of the ellipse (It can be any other points on major axis, but ...
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Are these family of involutes of $y^2=4x$
This relates to a very interesting MSE post:
How to solve $y'^2 +yy'+x=0$?
wherein the a family of curves orthogonal to the family of lines:
$$y=mx+1/m, m\in \Re \tag1$$
has been obtained as $$...
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Determing Obtuse or Acute Angle Bisector of Intersecting Straight Lines
I have two lines $$L_1: a_1x+b_1y+c_1=0$$ $$L_2: a_2x+b_2y+c_2=0$$ that are intersecting at some point and the equation of the line bisecting the intersecting angle is
$$\frac{a_1x+b_1y+c_1}{\sqrt{a_1^...
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Inscribed ellipse in a triangle that has a known orientation and a known tangent point
Given a triangle, you want to determine the inscribed ellipse in it, which has a known orientation, specified by a known rotation matrix, and in addition, one of its tangent points with the three ...
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Verifying $\delta$ in $\lim_{x\to 3}x^2=9$ for $\delta = \sqrt{9+\epsilon}-3$.
In Stewart's Calculus on page 81 question 34 it says the following:
Verify, by means of a geometric argument, that the largest possible choice of $\delta$ for showing that $\lim_{x\to 3}x^2=9$ is $\...
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Area enclosed by a hypocycloid with 5 cusps
Problem
Area enclosed by the hypocycloid with 5 cusps
$\left\{\begin{aligned}
x&=\frac35\cos(t)+\frac25\cos\left(\frac{3t}{2}\right)\\
y&=\frac35\sin(t)-\frac25\sin\left(\frac{3t}{2}\right)
\...
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Discriminant of a Real Polynomial vs Complex Polynomial
Suppose $f(x) = ax^3 + bx^2 + cx + d$, with $a,b,c,d \in \mathbb{R}$.
Let $\Delta$ denote the discriminant. If $\Delta > 0$, then $f$ has 3 distinct, real roots. If $\Delta = 0$, then $f$ has a ...
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