Questions on the use of algebraic techniques to prove geometric theorems.

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Parabola problem

Water squirting out of a horizontal nozzle held $4$ ft above the ground describes a parabolic curve with the vertex at the nozzle. If the stream of water drops $1$ ft in the first $10$ ft of ...
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Find the equation of the perpendicular bisector of AB for: [on hold]

Find the equation of the perpendicular bisector of AB for: A(1,3) and B(-3,5) What I did: m2= 3-5/1+3 multiply m2=-1 =1/2 MidPt:(3+5/2, 1-3/2) =4,-1 Equation of PB of AB is: Y=1/2x+-2 ???
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Deduce the inequalities $3\lt \pi \lt 12(2-\sqrt{3})$, by calculating the areas of regular twelve-sided polygons.

Calculate the areas of regular dodecagons (twelve-sided polygons) inscribed and circumscribed about a unit circular disk and thereby deduce the inequalities $3\lt \pi \lt 12(2-\sqrt{3})$. This is a ...
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A better way to answer this question

So my team and i were asked this question a few years ago on a small Math-A-Thon on my hometown. It went something like this: "We need to transport a neon tube (or any tube, who cares) of 92cm ...
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38 views

Find the Ratio $BM \colon ME$

In Triangle $\Delta ABC$, the Point $D$ is on $BC$ such that $D$ divides $B$ and $C$ in the Ratio $1 \colon 3$ and there is a point $E$ on $CA$ such that $E$ divides $C$ and $A$ in ratio $1 \colon 3$. ...
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Equation of the locus

Find the equation of the locus of a point $P = (x, y)$ when the sum of the squares of the distances from $P$ to the points $(a, 0)$ and $(-a, 0)$ is $4b^2$, where $b \geq \dfrac{a}{\sqrt{2}}$?
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How to get rid of the term with $xy$?

I'm trying to put this conic on an identifiable form. $$4x^2-4xy+y^2+20x+40y=0$$ I know that the term $xy$ implies that I need to rotate the conic so that $xy$ vanishes. But I've read on some books ...
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Should the expanded expression of a quadratic form be equals to It's original expression?

Sorry if the question is a little misleading, but I have no better way to express it. The text below should clarify. Suppose I have the equation of a conic: $x^2+y^2+z^2-2x+3y+z+2=0$, with this I ...
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1answer
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Point within a Cube in 3D environment

I have a cube in 3D space with 8 corner points with their X,Y,Z Coordinates. I know how to test if any given point lies inside a cube by Comparing their coordinates to be greater or smaller than the ...
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Start and end point of a rotated ellipse

I have the data of an incomplete ellipse and I need to retreive the minimun information in order to describe an elliptical arc. In particular following are my ellipse data: Major axis vector (x, y) ...
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Given a Line Parametrization, Finding another Equation

So I am given a line $l$ with the parameterization, $x=t, y=2t, z=3t$. Now let some point, $p$ be a plane that contains the line $l$ and the point $(2,2,2)$. So given this, how do I find an equation ...
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1answer
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Where are these choices of $A',B',C'$ for this quadratic form?

I'm studying quadratic forms: In the book I'm reading, he starts by looking at quadratic forms such as: $$\varphi (x,y)=Ax^2+2Bxy+Cy^2$$ And that given this quadratic form, one can introduce via ...
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2answers
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The Reason for different Forms of Equations

I recently started learning about conic sections and saw people writing the equations for the different figures (circle, parabola, ellipse, and hyperbola) in different forms. (standard form, vertex ...
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1answer
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For line $ax+by+k=0$ which intercepts form a triangle rectangle with area $A$, find $k$

I know that the area of a triangle is given by the formula $A=\frac{1}2Bh$ and the intercepts of line $ax+bx+k=0$ are $(B,0)$ and $(0,h)$ which forms a square with area $2A$, but without brute-forcing ...
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When should I shift $a$ and $b$ in $\cfrac{x^2}{a^2}+\cfrac{y^2}{b^2}=1$?

Find the reduced equation of the elypsis such that: The foci are $(0,6);(0,-6)$ and the larger axis has length $34$. I did the following: Taking the equation ...
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Question about the coordinates in a new origin on the plane.

I'm reading a book on analytic geometry, specifically on a chapter on change of coordinates. It says that having the origin $O$, one point $P$ and a new origin $O'$, the vector that describes the ...
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1answer
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A simplified formula for area of triangle when equations of the sides are given

For i = 1, 2, and 3, let $a_ix + b_iy + c_i = 0$ be three equations of 3 (non-special cased) straight lines. From which, the co-ordinates of the vertices can be found. Using these co-ordinates, via ...
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1answer
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Why does this hyperboloid change into a surface? [duplicate]

Given this equation $x^2+y^2+z^2+2xy+2xz+2yz-x-y-z=6$ and the corresponding quadric: If I rearrange the equation to $(x+y+z-3)(x+y+z+2)=0$ (which is equivalent), I get: So, which is the right ...
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1answer
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Points on two skew lines closest to one another

Given two skew lines defined by 2 points lying on them as $(\vec{x}_1,\vec{x}_2)$ and $(\vec{x}_3,\vec{x}_4)$. What are the vectors for the two points on the corrwsponding lines, distance between ...
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How to transform (rotate) this hyperbola?

Given this hyperbola $x_1^2-x_2^2=1$, how do I transform it into $y_1y_2=1$? When I factor the first equation I get $(x_1+x_2)(x_1-x_2)=1$, so I thought $y_1=(x_1+x_2)$ and $y_2=(x_1-x_2)$. ...
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Prove that a point can be found which is at the same distance from each of the four points$\ldots$

Prove that a point can be found which is at the same distance from each of the four points $\bigg(am_1,\dfrac{a}{m_1}\bigg),\bigg(am_2,\dfrac{a}{m_2}\bigg),\bigg(am_3,\dfrac{a}{m_3}\bigg)$ and ...
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Find the equation of base of Isosceles Traingle

Given the two Legs $AB$ and $AC$ of an Isosceles Traingle as $7x-y=3$ and $x-y+3=0$ Respectively. if area of $\Delta ABC$ is $5$ Square units, Find the Equation of the base $BC$ My Try: The ...
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1answer
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Area of Triangle Given 3 vertices

Given that $P=(1,1,0), Q=(1,0,1), R=(0,1,1)$. I need to find the area of the triangle. What I have done: I have tried finding the distances of PQ, QR, and PR. I have those distances, I don't know ...
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Co-ordinate geometry and area of triangle

When a straight line $ax+by+c=0$ forms a triangle with the axes $x$ and $y$, what is the general formula for the area of the triangle?
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What geometric object is given by this equation?

What geometric object is given by this equation? $x^2+y^2+z^2+2xy+2xz+2yz-x-y-z-6=0$ Maple says it's a hyperboloid of one sheet, but is there a way to show it without going the long way by using the ...
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Let $ S=\{(x,y)\in\mathbb{R}^2 \ | \ x^2+y^2=1 \text{ and } y\geq 0\}$. Determine $S+S+…+S $.

Let $$ S=\{(x,y)\in\mathbb{R}^2 \ | \ x^2+y^2=1 \text{ and } y\geq 0\}$$ By the usual notation for sum of sets let $$ 2S\overset{\text{not}}{=}S+S=\{(x_1+x_2,y_1+y_2) \ | \ (x_1,y_1), ...
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Is the focal distance of $x^2+\cfrac{2y^2}{3}=8$ equal to $2\sqrt{-4}$?

I've just computed the focal distance of $x^2+\cfrac{2y^2}{3}=8$ and found $2\sqrt{-4}=4i$ as follows: $$x^2+\cfrac{2y^2}{3}=8\\ \cfrac{x^2}{8}+\cfrac{y^2}{12}=1$$ Then the focal distance should be ...
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Is this solution legal?

Let $M(1,-1)$ be a point in a plane. Find its distance from a line given by $x+2y-4=0$. Later on I found a formula: $$d=\frac{\left | Ax_{0}+Bx_{0}+C \right | }{\sqrt{A^2+B^2}}$$ But I did it ...
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Algorithm to calculate line segments between two points bounded by multiple surfaces

Problem statement: As a specific case, let's say I have a volume composed of a series of concentric cylinders. Given a fixed point P (a,b,c), and another randomly sampled point Q (x0,y0,z0), I would ...
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Proof regarding hyperbolas

Given the parameters $a,b>0$ we set $c:=\sqrt{a^2+b^2}$ and $e:=\large\frac{c}{a}$ (eccentricity), the focal points are $F=(c,0)$ and $F'=(-c,0)$, the directrix $L$ with the equation ...
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3answers
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Finding extrema.

Find the minimum distance between point $M(0,-2)$ and points $(x,y)$ such that: $y=\frac{16}{\sqrt{3}\,x^{3}}-2$ for $x>0$ . I used the formula for distance between two points in a plane to get: ...
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distance between planes in a simplex

In an Euclidean space there are n points at equal distances d to each other (regular simplex). Find out a distance between two parallel planes, one spanned at points numbered 1 through k, the other at ...
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4-dimensional simplex

In a 4-dimensional Euclidean space, there is a simplex, with given lengths of all the edges aij = distance(Ai,Aj). Find a distance between gravity centers of sides, opposite to each other. Notice: ...
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Why $\|X-F\|=e|(X-F)\cdot N -d|$ should be written as $\|X-F\|=e|(X-F)\cdot N +d|$?

I'm reading Apostol's Calculus. $\quad $ And I've tried to do the following exercise: $\quad \quad \quad \quad $ I am a little confused: I have the portuguese version of the book, and it ...
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Help with Apostol's “Calculus, vol. 1”, Section 1.18

In section 1.18 ("The area of an ordinate set expressed as an integral"), Apostol proves two theorems. the first, theorem 1.10, deals with the area of a function's ordinate set; the second, theorem ...
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Definition of a Cartesian coordinate system

Apologies if this is a basic question, but I'd really like to clarify the exact meaning of what a Cartesian coordinate system is. Heuristically, is it correct to say that a Cartesian coordinate system ...
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1answer
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Graphical details of changing functions

I'm struggeling a bit with the transformations of a function when values are changed (for instance an offset to the right etc). So far I have found the following: ...
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Find the equations of the lines tangent to the circle $x^2+y^2=r^2$ that pass through the point $(a,0)$?

Find the equations of the lines tangent to the circle $x^2+y^2=r^2$ that pass through the point $(a,0)$. My book explains that the equation of this line is $y=m(x-a)$ and then we make the ...
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Questions on the relation of the axis of a cone to its conic sections

(1) Does the axis of a cone pass through the foci of any its conic sections? Consider the image below: Is the intersection of the axis of cone and the ellipse the same as the focus of the ellipse? ...
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Analytic geometry line segments

This is a very interesting analytic geometry math problem that I came across in an old textbook of mine. It is quite nice and I decided I would share it with MSE for future reference and a fun time?! ...
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Why the distance from the point to the line is $\frac{|(P-Q)\cdot N|}{\|N\|}$?

$P$ is a point in the line $L$, $N$ is a vector normal to $L$ and $Q$ is a point out of the line. I know that taking the subtraction of $(P-Q)$, I create a vector that goes from $P$ to $Q$ but I don't ...
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Is the following a conic section

All vectors are in $\mathbb{R}^3$ and only $\mathbf{r} = \left[ x; y; z \right]$ is unknown. My question is does the following system define a conic section in the $x-y$ plane and, if so, how can I ...
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Location of an arbitrary point of an ellipse

Given this ellipse equation $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$, $(a>b>0)$ and $c:=\sqrt{a^2-b^2}$ aswell as the focal points $F=(c,0)$ and $F'=(-c,0)$, why can we say without loss of ...
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Parametric equations of perpendicular lines

I'm having problems with this: Find the parametric equation of the line that passes through the point $(-1, 4, 5)$ and is perpendicular to the line: $$x = -2 + t$$ $$y = 1 - t$$ $$z = 1 + 2t$$
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How many sets of four points in an MxN grid have one point contained by three other points?

Given a 3x3 grid: 1 2 3 8 9 4 7 6 5 We find 126 distinct sets of 4 points $$\binom{9}{4}$$ There are 8 cases such that when the points are connected with a line in clockwise direction, one point ...
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3answers
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Finding the locus of midpoint of $AB$

The normal to the ellipse $b^2x^2+a^2y^2=a^2b^2$ is passing through the x-axis in point $A$ and through the Y-axis in point $B$. Point $P$ is the midpoint of $AB$. Need to find the locus of $P$. ...
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1answer
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Connected component identification?

Suppose I give a random 2 variable polynomial relation such as: $$x^3+y^3=10$$ $$x^2 + 7yx^4 + x^2-15=0$$ Etc... How do I determine how many individual pieces there are to the graph?
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Solving euclidean geometry problems with analytical geometry

Can anyone recommend a good resource about applications of analytical geometry in doing elementary geometry problems like ones on IMO?
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28 views

Identity simplification

How do you express $\dfrac{\sin A\sec A\cot A}{\tan A}$ in terms of sine and cosine? I have simplified using $\sec(A)$ as $\cos^{-1}(A)$ and also $\cot(A)$ as $\dfrac{\cos(A)}{\sin(A)}$, and appear ...
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What's interesting in latus rectum?

I'm a maths teacher in Italian secondary school and I've been spending some time trying to construct "meaningful" problems about conic sections. I particularly like problems which focus on practical ...