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

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Are all curves with equation of the form $(\xi x +n) \cdot x = \text{const}$ circles?

Let $x(t)=(x_1(t),x_2(t))$ with $t\in [a,b]$ be a smooth curve in $\mathbb{R}^2$ and $\xi \in \mathbb{R}$ such that $$(\xi x +n) \cdot x = \text{const}$$ Here $n$ is the unit normal to the curve. Is ...
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15 views

locus of a variable straight line [on hold]

Geometry: A variable straight line always intersects the lines x=c,y=0; y=c,z=0; z=c,x=0. find the equation to its locus. taking the equation of a line in parametric form and substitute the given ...
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18 views

Projection of a Vector on a Straight Line in $\mathbb{R}^3$

I have the following: Consider the straight line $(\epsilon)$ which passes through the origin and forms an angle $t$ with $Ox$ axis. Find the matrix $A$ which projects a random vector ...
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1answer
17 views

Finding $y_{A,i}$ and $y_{B,i}$ in this geometric relationship problem.

Finding $y_{A,i}$ and $y_{B,i}$ in this geometric relationship problem. I'm an high-speed aerodynamics student. I am studying a sweptback wing like in the figure below (in green). Notice that I ...
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Equation of plane perpendicular to given plane

Find the equation of the plane which contains the line of intersection of the planes $x+2y+3z-4=0$ and $2x+y-z+5=0$ and which is perpendicular to the plane $5x+3y-6z+8=0$ By setting $z=0$ I found a ...
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1answer
17 views

Equation of line passing through origin

Find the equations of the two lines through the origin which intersect the line $\frac{x-3}{2}=\frac{y-3}{1}=\frac{z}{1}$ at angles of $\frac{\pi}{3}$ Now our required line should be ...
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54 views

Perimeter of a teardrop (made by two adjacent circles)

I'm trying to determine the perimeter of a teardrop shape formed by two adjacent circles (non-intersecting) with mutually tangent lines drawn on both sides of the circles. I've attached a sample ...
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20 views
+100

Trigonometric position function and intersection

I have the following position function for a point: $x(t) := C_x - (S_x-C_x) \cdot \cos(t\cdot\theta) + (S_y-C_y) \cdot \sin(t\cdot\theta) + t \cdot v_x$ $y(t) := C_y - (S_x-C_x) \cdot ...
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3answers
31 views

Find the condition such that one of the lines defined by $ax^2+2hxy+by^2=0$ has slope $k$ times that of the other

Find the condition that the lines represented by $$ax^2+2hxy+by^2=0$$ are such that the slope of one line is $k$ times that of the other. I calculated the two represented by $ax^2+2hxy+by^2=0$ ...
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1answer
31 views

Expressing a point in $\mathbb{R}^n$ as a sum of unit vectors

I'm pretty sure that any point in $\mathbb{R}^n$ can be written as a sum of finitely many unit vectors (in $\mathbb{R}^n$, of course). However, I have no idea how to go about proving this. Any ideas? ...
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1answer
16 views

Proving the square formed by pairs of lines

Show that the two pairs of lines $12x^2+7xy-12y^2=0, 12x^2+7xy-12y^2-x+7y-1=0$ form a square. I know that both the equations represent a pair of straight lines. Also the first equation represents a ...
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1answer
20 views

Finding mid-point of $BC$ if point $A$, orthocenter and circumcenter are given in a triangle

If in a triangle $ABC$, $A \equiv (1,10)$, circumcenter $\equiv (-\frac13, \frac23)$ and orthocenter $\equiv (\frac{11}3, \frac43)$ then the coordinates of mid-point of side opposite to A is? ...
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1answer
21 views

Equation of a Pair of Straight Lines .2nd degree

Show that if one of the lines given by $a_1x^2+2h_1xy+b_1y^2=0$ coincides with one of the lines of $a_2x^2+2h_2xy+b_2y^2=0$ then $(a_1b_2 - a_2b_1)^2=4(a_2h_1 - a_1h_2)(b_1h_2-b_2h_1)$ Actually, I ...
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1answer
42 views

Proving equilateral triangle

Show that the lines $x^2+16xy-11y^2=0$ form an equilateral triangle with the line $2x+y+1=0$ and find its area. --------________________________--------- My solution is here; Here $x^2+16xy-11y^2=0$ ...
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1answer
28 views

No. of points determining a unique parabola

For a parabola, let Focus: $(a_1,b_1)$ Equation of directrix: $y-mx-c=0$ The equation of parabola is, $\sqrt{(x-a_1)^2+(y-b_1)^2}= \frac{|y-mx-c|}{\sqrt{1+m^2}}$ There are 4 parameters ...
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27 views

Finding the equation of diagonal

If $ax^2+2hxy+by^2=0$ be the two sides of a parallelogram and $px+qy=1$ is one diagonal then prove that the other diagonal is $y(bp-hq)=x(aq-hp)$. My solution is here; $ax^2+2hxy+by^2=0$ Multiplying ...
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5answers
60 views

Calculating the area of Triangle

Find the area of triangle formed by the lines $x^2+4xy+y^2=0$ and $x+y=1$. I know that the equation $x^2+4xy+y^2=0$ represents a pair of straight lines but how do i factorize it to get the two lines ...
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31 views

Equation of a plane given one point and two planes

I've done a question similar to this, however this one has no complete equations i can solve for. Determine the equation of the plane that passes through $(1,3,8)$ and is perpendicular to the line ...
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1answer
31 views

Find sides of a right triangle given hypotenuse c and area A (no numbers given)

I've solved couple of these, but I have no idea how to solve it without any numbers provided. I've tried using $A=\frac{ab}{2} \Rightarrow 2A=ab \Rightarrow 4A^2=a^2b^2$ and incorporating ...
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1answer
45 views

Question in the proof of the Brower fix point theorem

One can show that for any given homology theory $H$ with non-trivial coefficient group $G$ there does not exist a retract $\partial B^n \subset B^n$. Brower's fix point theorem states that any ...
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3answers
30 views

Are certain equations for orthogonal trajectories of a curve incomplete?

Suppose we wish to observe a Euclidean circle $C$ with radius $\alpha$. We define the relation $$R=\{(x,y)\in\mathbb{R}^2:\alpha^2=x^2+y^2,\text{fixed}\,\alpha\in\mathbb{R}^{+}\},$$ represented in ...
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18 views

How to assign variables to points on the cartesian plane?

Suppose I want to name the point $(3,4)$ a name say A. Can I just say $A=(3,4)$?
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1answer
22 views

equation of straight lines through a point which divide the circumference of a circle in the ratio $2:1$ [closed]

Find the equations of straight lines which pass through the intersection of the lines $x-2y-5=0$ and $7x+y=50$ and divide the circumference of the circle $x^2+y^2=100$ into two arcs whose lengths are ...
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1answer
26 views

Equation of a circle whose radius and tangent is given

Equation of a circle which passes through the origin, whose radius is $a$ and for which $y = mx$ is a tangent.
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5 views

Why is convenient to consider points conjugate axis in hyperbola?

In "The Concise Oxford Dictionary of Mathematics" I found this: It may be convenient to consider the points $(0,-b),(0,b)$ on the conjugate axis, despite the fact that the hyperbola doesn't cut ...
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1answer
33 views

Rational points on a line

This question is quite unique. Does there exist some point in the coordinate system such that any line passing through it has at most 2 rational points lying on it?
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1answer
27 views

How can I find the location on the Z axis where two skew lines pass closest to each other on the XY plane?

I'm given one point $\overrightarrow{P}$ and the slopes $\frac{dX}{dZ}$ and $\frac{dY}{dZ}$ for each of my two lines. I'm trying to figure out where on the Z axis my lines pass closest to each ...
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1answer
31 views

Geodesic distance between equidistant points on a sphere [closed]

On the unit sphere equidistant points can be found for $1, 2, 3, 4, 6, 8, 12, 20$. The geodesic distance between the points are $\pi$ for $2$, $2\pi\over 3$ for $3$, $\pi\over 2$ for $6$, etc... Is ...
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1answer
45 views

What is the equation of a pyramid with a square base?

Which algebraic description can be found for a pyramid, defined as a scalar function $$f:\mathbb{R}^2 \rightarrow \mathbb{R}$$ $$(x,y)\rightarrow z$$ Particular assumptions: Square base $z=0 \iff ...
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2answers
20 views

line parallel to x-axis and arbitrary intersection test

I am going through code snippets that calculate the x-intersection point between the line parallel to the x-ais and an arbitrary line between points (x1,y1) and (x2,y2). The code snippet does the ...
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1answer
29 views

General equation for a line contained in a plane and passing through a point

I have a vector $n$ and I seek a parametric equation for a line that is orthogonal to $n$ and passes through a point $(a,b,c)$. I got the equation of the plane formed by the normal vector and that ...
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1answer
31 views

Line equation through point, parallel to plane and intersecting line

Write the equations of the line that passes through point $M(1,0,7)$, is parallel with the plane $3x-y+2z-15=0$ and intersects line $\frac{x-1}{4}=\frac{y-3}{2}=\frac{z}{1}$ Alright, so from what I ...
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2answers
29 views

Height of a paralelogramm

I have the coordinates of the 4 vertexes of a parallelogram. If i calculate the length of two opposing sides, how do I get the perpendicular distance between them? Is it just the distance between the ...
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2answers
41 views

If a matrix $\boldsymbol{\mathrm {A}}$ is orthogonal, its determinant is $\pm 1$. Is the converse also true?

I know that an orthogonal matrix satisfies $$~~~~~~~~~~~~~~~~~~~~~~~~~\boldsymbol{\mathrm {AA}}^T=\boldsymbol{\mathrm {A}}^T\boldsymbol{\mathrm {A}}=\boldsymbol{\mathrm {I}}~~~~~~~~~~~~~~~~~~~~~~(*)$$ ...
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1answer
57 views

Triangle with same black and white areas

Suppose we have an infinite chessboard with the usual black/white coloring. A triangle $T$ with area $a$ is given with vertices at corners of some cells. Prove that there exists another triangle $T'$ ...
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1answer
28 views

Locus of vertex

A variable parabola of latus rectum $l $, touches a fixed equal parabola , the axes of the two curves being parallel . Then locus of the vertex of the moving curve is a parabola , then what is the ...
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2answers
48 views

Number of inscribed triangles in a rectangular hyperbola touching a parabola

How many triangles can be incribed in the rectangular hyperbola $xy= c^2$ whose sides all touch the parabola $y^2 =4ax$. How can we start the question . Please help.
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38 views

Classify the set $\{(x,y,z):f(x,y,z)=0, \nabla f=0\}$, where $f$ is a polynomial of degree at most 3.

Suppose that $f(x,y)$ is a nonzero polynomial of degree at most 2. Observe the following set: $$S=\{(x,y) : f(x,y)=0 ,\; \partial_{x}f(x,y)=0,\; \partial_{y}f(x,y)=0 \}.$$ Note that this set is the ...
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2answers
48 views

Prob. 19, Chap. 1 in Baby Rudin: For what $\mathbf{c}$ and $r > 0$ does this equivalence hold?

Here's Prob. 19 in Chap. 1 in the book Principles of Mathematical Analysis by Walter Rudin, 3rd edition: Suppose $\mathbf{a} \in \mathbb{R}^k$, $\mathbf{b} \in \mathbb{R}^k$. Find $\mathbf{c} \in ...
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Prob. 16, Chap. 1 in Baby Rudin

Here is Prob. 16, Chap. 1 in the book Principles of Mathematical Analysis by Walter Rudin, 3rd edition: Suppose $k \geq 3$, $\mathbf{x}, \mathbf{y} \in \mathbb{R}^k$, $\vert \mathbf{x} - ...
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1answer
48 views

Proof of equal angles in a quadrilateral.

points E and F are given on side BC of a convex quadrilateral ABCD (with E closer than F to B). Suppose angle EAB = angle CDF and angle FAE = angle FDE. Prove that angle CAF = angle EDB.
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4answers
50 views

Equation of a circle tangent to two lines , given the radius . [closed]

What is the equation of the circle whose center is in the first quadrant and with the radius of $4$ units, given that it is tangent to the $x$-axis and to the line $4x-3y=0$?
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2answers
48 views

How to calculate one of the vectors that generate a given cross-product?

Given the vector: $$\vec b=(-0.361728, 0.116631, 0.924960)$$ and it's cross-product: $$\vec a \times \vec b=(-0.877913, 0.291252, -0.380054)$$ How do I calculate $\vec a$ ? It's been a while since ...
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3answers
25 views

Reflecting coordinates over the line $x = -1$

I know how to reflect a coordinate over the $y$ and $x$ axis, but is there a rule I could use to help me find the reflected point over $x = -1$? This is what I know already: Over the $x$-axis: ...
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2answers
31 views

General form of intersection line between 2D plane and 3D hyperboloid when offset from symmetry axis?

General Background Suppose I have one side of a two-sheet hyperboloid as a general three dimensional shape, where the symmetry axis is along, say, my x-axis ($\hat{\mathbf{x}}$) in my chosen ...
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1answer
30 views

remove the interior points of two intersected closed-curves

The problem is as follows I have two intersected closed curves and each curve was represented by two arrays respectively, which means we know the coordinates of every points $(x_i,y_i)$ but no ...
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2answers
43 views

How can one measure distance between point and the line in maximum metric space?

Given metric space $M = (\mathbb{R}^2, d)$ where $d = \operatorname{max}\{|x_1 - y_1|, |x_2 - y_2|\}$, how can one measure distance from some arbitrary point $X$ to the line $y = 3$, let's say? How ...
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1answer
6 views

Finding the equation of a coaxial circle with its diameter falls on the radical line

Here is the problem:- $L: x – y + 3 = 0$ is the radical line for $S$, the system of coaxial circles. $C: x^2 + y^2 – 2x – 4y – 11 = 0$ is a member of $S$ with $AB$ as the common chord. (a) Find the ...
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2answers
46 views

Tangent line of Lissajous curve?

I'm trying to find at how many points the tangent line of $(\cos(3t),\sin(2t))$ goes through the point $(3,0)$. My attempt: This is the same thing as saying for how many values of $t$ do we have ...
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1answer
45 views

The tangent at a point $P$ on the curve $y=\ln(\frac{2+\sqrt{4-x^2}}{2-\sqrt{4-x^2}})-\sqrt{4-x^2}$ meets the $y-$axis at $T,$then find $PT^2.$

The tangent at a point $P$ on the curve $y=\ln(\frac{2+\sqrt{4-x^2}}{2-\sqrt{4-x^2}})-\sqrt{4-x^2}$ meets the $y-$axis at $T,$then find $PT^2.$ Let the point of tangency be $P(x_0,y_0)$ on the ...