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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15 views

Affine transformation between compact sets

So, I am starting mysefl on the study of the Finite Element Method for numerical PDE and I have found the deffinition of "affine equivalent methods", that is, that two FEM, $(K,P,\Sigma)$ and $(K^*,P^*...
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21 views

What is the homogenous parametric equation of a line given two points?

In the case we have two non-homogeneous points $\vec{A}=(x_1,x_2)$ and $\vec{B}=(y_1,y_2)$, we can write the vector equation of the line with a real parameter $\lambda$: $r: \quad \vec{X}=\vec{A}+\...
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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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25 views

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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35 views

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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26 views

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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Existence of $f$ such that $f(x,|x|^2)f(y,|y|^2)=0$ whenever $x \cdot y=0$

Does there exists a non trivial continuous function (other than $f=0$) with the following : $f:R^4 \to [0, \infty)$ Let a $x,y \in R^3$ and their respective Euclidean norm squared $|x|^2$ and $|y|^...
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Prove interval of x-y for a circle quation

Let $x, y \in \mathbb{R}$ such that $x^2 + y^2 - 4x + 10y + 20 = 0$. Prove that $y + 7 - 3\sqrt{2} \le x \le y + 7 + 3\sqrt{2}$. I'm struggling with that problem. I've recognized that the given ...
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48 views

Solution verification: given the point $T$ and the line $p$, find the distance between their orthogonal projections onto the plane $\pi$.

Let $T'$ and $p'$ be the orthogonal projections of the point $T(-8,2,-3)$ and the line $$p\ldots\frac{x}4=\frac{y-4}3=\frac{z+1}{-2}$$ onto the plane $\pi\ldots x-y+3z+8=0$. Find the distance ...
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IMO complex number geometry problems

I've been trying to master complex number geometry for some time and now I'm having a hard time finding problems suitable for complex numbers. Can anyone suggest some IMO or other olympiad problem ...
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What is Semi-diameter of a conicoid?

My University book is really confusing about this topic , can somebody explain and also show a visual representation of it if possible?
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21 views

Shadow rushed by a disk in space

A disk of radius 1 is centered at the point $A(0,1,2)$ and is parallel to the plane $xOy$. A source of light is placed at the point $P(0,1,4)$. Characterize analytically the shadow and the disk rushed ...
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135 views

Prove: Three tangents to a parabola form a triangle with an orthocenter on the directrix and a circumcircle passing through the focus

Prove the following: The intersection points of any three tangents to a parabola given by the formula $y(y-y_0)=2p(x-x_0)$ are vertices of a triangle whose orthocenter belongs to the directrix ...
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24 views

Find the proportion of the length $PQ$ and $PR$ in the following triangle.

Given triangle $ABC$ with right angle $A$ and $D$ is the midpoint of $BC$. $F$ divides $AB$ into two equal parts and $E$ and $G$ divides $AF$ and $FB$ into two equal parts, respectively. The line $AD$ ...
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25 views

Parabola transformation

Find the real affine change of coordinates that maps the parabola in the $xy$-plane to the parabola in the $uv$-plane $$4x^2 + 4xy + y^2 - y + 1 = 0$$ $$4u^2 + v = 0$$ My attempt: Since there is an $...
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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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80 views

Is there a formula for bounded area between line and parabola?

The area (in sq. units) of the region described by $A=\{(x,y): \frac{y^2}{2} \leq x \leq y+4\}$ is $18$. The area (in sq. units) of the region described by$A=\{(x,y): y^2 \leq 2x \; \textrm{and} \; ...
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Distance between two generated lines

Write the equations of the rectilinear generatrices of the hyperbolic paraboloid $\frac{x^2}{p}-\frac{y^2}{q}=2z\ \ ,p,q>0$. Out of these rectilinear generatrices select those which are parallel to ...
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Sine difference identity

I'm trying to prove that given 2 vectors $\vec{a} = A(\cos{\alpha}, \sin{\alpha}) $ and $\vec{b} = B(\cos{\beta}, \sin{\beta}) $ the following relation is true by using the exterior product with the ...
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51 views

Form a new base with two vectors

I need to make a new orthonormal base in $\mathbb{R^3}$ given $(2,7,5)$ and $(4,1,3)$ so that it makes $(\widehat{e_1}, \widehat{e_2}, \widehat{e_3} )$. But $ \widehat{e_1} $ has the same direction of ...
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Intersection of the hyperboloid with its tangent plane

Find the intersection of the hyperboloid $x^2+y^2-3z^2=1$ with its tangent plane at the point $(2,0,1).$ I know the equation of the tangent plane at that point is $2x-3z=1$. But isn't the ...
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Injective function from unit circle [closed]

Let $S$ denote the set of points on the unit circle centred at $(0,0)$. Does there exist an injective function $f : S \rightarrow S \setminus \{(1,0)\}$?
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23 views

Finding the equation of a line passing through point of intersection

Find the equation of the line that passes through the point of intersection of $3x-5y+10=0$ and $2x+3y=6$ and also passes through the point $(-2,0)$. I have an idea on how to do this but i'm not sure ...
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Find vertex coordinates of a square given their distances $p,\>s,\>q,\>r$ to an inner point [closed]

The figure represents a square, with P a point inside the square. The four segments are drawn from P to the four vertices of the square and they are named p, q, r, s. If the bottom-left vertex is at ...
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1answer
42 views

Unique descompisition of a function as a sum of an isometry and a bounded function.

I have the following question: Let $h \colon \mathbb{R}^n \rightarrow \mathbb{R}^n$ such that $h(x)=\Psi(x)+\rho(x)$ with $\Psi$ isometry and $\rho$ a bounded function. Are $\Psi$ and $\rho$ unique in ...
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57 views

Find the equation of the cylinder whose two directions are given.

The radial directions of a cylinder is given by $$ \begin{array} wx^2+y^2=5^2, \\ z=0 . \end{array} $$ and axial directions of a cylinder is $$\vec a=(5,3,2)$$ respectively. Find the equation of ...
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How many parameters does the set of all spheres, which satisfy the given condition, depend on?

How many parameters does the set of all spheres, which satisfy the given condition, depend on? (i) Spheres that pass through the given point. (ii) Spheres that touch the given line (iii) ...
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Direction Cosines of three mutually perpendicular lines form an orthogonal matrix

I was going through a proof about a result in spheres (the sum of squares of intercepts of 3 mutually perpendicular lines from the same point to a sphere is constant), and the direction cosines of the ...
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Find the equation of the ellipse whose symmetry axes are given by $x+y-2=0$ and $y-x-1=0$. Also semi width $a=2$ and semi height $b=1$.

Find the equation of the ellipse whose symmetry axes are given by $x+y-2=0$ and $y-x-1=0$. Also semi width $a=2$ and semi height $b=1$. As the center can be found at the intersection of the symmetry ...
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1answer
59 views

Prove that A',B',C' are in a straight line.

Be an ABC triangle and a point P of your plane. The perpendiculars to PA, PB and PC, traced by P, intersect the BC, CA and AB sides at three points, A ', B' and C '. Prove that these points are in a ...
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59 views

Write the equation of a second-order line $2x^2-12xy-7y^2+8x+6y=0$ in canonical form

Write the equation of a second-order line $$2x^2-12xy-7y^2+8x+6y=0$$ in canonical form. To find its center, we have the following system of equations: $$2x-6y+4=0, \\ -6x-7y+3=0.$$ Hence, the center ...
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1answer
31 views

A mapping that converts a line segment to another one

I'm looking for a mapping $f:\mathbb{R}^2\to\mathbb{R}^2$ that converts a line segment $AA'$ with two end points $A=(x_A,y_A)$ and $A=(x_{A'},y_{A'})$ to another line segment $BB'$ with end points $B=(...
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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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51 views

The Number of Hyperplanes Intersecting a Unit Hypercube

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| + ....
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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 ...
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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 ...
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52 views

Recommend book on elementary geometry

I am seeking recommendations for books on elementary geometry, including Euclidean geometry and analytic geometry.
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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)...
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68 views

How do you mathematically characterize an “enlarged probability simplex”?

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 ...
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2answers
17 views

Proof involving projection

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{...

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