# 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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### Peas inside a pod [duplicate]

Question taken from (page 102) https://drive.google.com/file/d/1vfZ9vcFjNVeuv25wlCmSUM_AaWq6SNvN/view I am not able to understand the solution provided in the pdf. I tried using the equation of normal ...
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### Locus of a moving point, when constraints on an angle and length are given

$APQ$ is a variable triangle; $A$ is fixed, $P$ moves on a fixed line $CD$; if $AP$ meets a fixed line parallel to $CD$ at $R$, and if $PQ=AR$ and if the angle $APQ$ is constant, prove that the locus ...
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### Locus of a moving point, such that two distances have a common ratio

A, B are two fixed points on a fixed circle; P is a variable point on the circle; Q is a point on BP, such that BQ/AP is constant; find the locus of Q. The only approach I could think of is through ...
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### Hyperboloid of one sheet

As an engineer at Ghana Atomic Energy, you have been tasked to design a cooling tower in the shape of hyperboloid of one sheet. The horizontal cross sections of the cooling tower are circular with 10m....
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### Existence of $f$ such that $f(x,|x|^2)f(y,|y|^2)=0$ whenever $x \cdot y=0$ with $f(x,|x|^2)f(y,|y|^2)=g(x+y,|x|^2+|y|^2)$

This is based on my previous question see here but with additional requirements Does there exists a non trivial continuous function (other than $f=0$) with the following : $f:R^4 \to [0, \infty)$ ...
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### Find the equation of the chord of the ellipsoid that passes through $M(2,1,-1)$ and is divided equally with this point

Given an ellipsoid with the formula: $$\mathcal{I}: \ \ \frac{x^2}{25}+\frac{y^2}{16}+\frac{z^2}{9}=1$$ Find the equation of the chord that passes through $M(2,1,-1)$ and is divided equally with ...
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### Find the slope of the line intersecting a parabola [closed]

While doing my quarantine package,a came across the following question and tried to find area of the triangle using the determinant formula of triangle, but it didn't work. So I would like to have ...
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### Find equation of non-concurrent curve

A parabola and straight line (red) $$y-\frac{x^2}{2}-1=0,\quad y-\frac{x}{2}-2=0;\, \tag 1$$ are combined plotted and found to intersect at $P(-1,1.5),Q(2,3)\;;$ Two more curves (blue) are ...
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### Calculus: Early Transcendentals, 7th ed(stewart)-chapter 12 problems plus exercise 3

I tried to resolve the following excercise but i got stuck. 3)Let be $L$ the line of intersection of the planes $cx+y+z=c$ and $x-cy+cz=-1$, where $c$ is a real number. (a) Find symmetric equations ...
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### Look at equation $6xy - 30x + 20y - 100 = 0$. Do transformation coordinate axes to change the cone section to standard form! [closed]

The full question: Look at equation $6xy - 30x + 20y - 100 = 0$. Do transformation coordinate axes to change the cone section to standard form? Sketch out the cone sections graph Its my task from my ...
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### What is the center most point of a set of points?

Given a set of points in the plane $\\{p_1, p_2, \cdot \cdot \cdot, p_n\\}$ from which I don't know their position but the distances from each other. How can I determine the point that lies closest to ...
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Prove that the number of hyperplanes such that $$c_1x_1 + c_2 x_2 + ... + c_n x_n = 0, \pm 1, \pm 2, \pm 3, ...$$ which intersect the unit $n$-cube, $0< x_i < 1,$ is at most $$|c_1| + |c_2| + .... 1answer 42 views ### Conversion from one mathematical form to another. I was trying to understand a solution, when a encountered this line:$$or,\;\sum(a+\lambda l)^2=\lambda^2(l^2+m^2+n^2)\\[10pt]or,\;\lambda=-\frac{a^2+b^2+c^2}{2(al+bm+cn)}$$I tried various method ... 3answers 37 views ### Find common points of (x^2+y^2)^2=8xy  and (x-1)^2+(y-1)^2=1 Im trying to count an area limited by these two curves, but one step of my solution needs to find their intersection points, so i can find angle which i have to put to polar coordinates. When i start ... 1answer 52 views ### Recommend book on elementary geometry I am seeking recommendations for books on elementary geometry, including Euclidean geometry and analytic geometry. 2answers 126 views ### locus problem in analytical geometry asking about a constant sum of two tangents to two identical circles yielding an ellipse You are given two circles: Circle G: (x-3)^2 + y^2 = 9 Circle H: (x+3)^2 + y^2 = 9 Two lines that are tangents to the circles at point A and B respectively intersect at a point P such ... 0answers 29 views ### Find the minimum value of k such that \sum_{i=1}^5 (PP_i)^2 = k, P_i = (r,r^2) P is a point on the coordinate plane. Find the minimum value of k such that$$\sum_{i=1}^5 (PP_i)^2 = k$$where P_i = (i,i^2). (PP_i) denotes the distance between point P and point P_i This ... 1answer 27 views ### Geometric sequence not correct? So I was checking some Khan Academy excercise about a sequence and it went something like this...$$4, 25, 100...$$It said that f(1)=4, f(2)=25 and f(3)=f(1)f(2). So I was thinking about how ... 0answers 34 views ### Trisecting an angle with a compass and 2 marks on a ruler It's well known that there is no possibility to trisect and angle with a compass and a ruler. But there is such a procedure when the ruler has 2 marks on it. See the snippet below. It's not very ... 4answers 60 views ### Find the asymptotes to the hyperbola 3x^2+2xy-y^2+8x+10y+14=0 The hyperbola is given with the following equation:$$3x^2+2xy-y^2+8x+10y+14=0$$Find the asymptotes of this hyperbola. (\textit{Answer: } 6x-2y+5=0 and 2x+2y-1=0) In my book, it is said that ... 2answers 29 views ### Find the equation of all circles tangential to the lines y = 0, x = 0 and y = - x + 2 I have a question, to find the equation of all circles tangential to the lines y=0,\,x=0 and y=-x+2. There should be 4 circles. I understand so far that circles take the form$$(x - h)^2+(y - k)...
We all know that the probability simplex can be described as the set $$\Delta = \left\{\theta \in \mathbb{R}^n| \sum\limits_{i = 1}^N \theta_i = 1, \theta_i \geq 0\right\}$$ and in $\mathbb{R}^3$ it ...
I need to prove the following: If $\vec{A}$ and $\vec{B}$ are two vectors different from $\vec{0}$. Proof $\vec{A} - c \vec{B}$ is orthogonal to $\vec{B}$ if \$ c = \frac{\vec{A} · \vec{B} }{||\vec{...