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

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0
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1answer
27 views

2D coordinates of rotating a “bent line”?

I have this problem, when I am given a point A an an XY plane, and I need to find the coordinates of a point B that is of a constant distance of my point A, and my OAB angle is fixed (O being the ...
0
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1answer
24 views

A question about affine spaces

Are there affine spaces that contain subsets that aren't closed to affine combinations of three points? This is a surprising question. I think that exists that kind of affine spaces,but I don't know ...
2
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3answers
33 views

$P$ is a point on ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ $(a>b)$ and $S$ and $S'$ are its focii

If $P$ is a point on ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ $(a>b)$ and $S$ and $S'$ are its focii. $\angle PSS'=\alpha$ and $\angle PS'S=\beta$, then prove that: $$ ...
3
votes
2answers
50 views

Best Fitting Pipe in parabolic trench

A work crew is digging a pipeline. The cross section of the trench is in the shape of the parabola $y = x^2$. The pipe has a circular cross section. If the pipe is too large, then the pipe will not ...
0
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1answer
17 views

find the coordinates of the point that divides the join of A(-1,-7) & B(1,2) internally, in 2:1.

What I wanted to ask was that after finding the coordinates of the point my answer was (1/3, -1) now since the ordinate is -ve doesn't that make this an external division? How can it divide the line ...
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4answers
56 views

Coordinate Geometry: Are there enough information to find out the coordinates?

Question: Given the circle $x^2+y^2=25$ is inscribed in triangle $\triangle ABC$, where vertex $B$ lies on the first quadrant. Slope of $AB$ is $\sqrt 3$ and has a positive y-coordinate, and ...
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2answers
28 views

Rotation of conics [duplicate]

How to rotate a conic by an determined angle? Could someone give me the step by step? (I know how to rotate the coordinate system by that formula \begin{align} x &= x'\cos(a) - y'\sin(a) \\ y ...
3
votes
1answer
59 views

A mirror focusing beams at one point

How can I find a shape of a mirror which focuses all parallel beams in one point? I tried to do it in this way: The mirror must be symmetric hence I assumed it has a center in the point $(0,0)$. The ...
0
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2answers
72 views

Get the four corners of a rectangle

I have a boundary given ($xMin$, $yMin$, $xMax$, $yMax$) and the two points of a reference line of a rectangle. The begin point is at $(x_b, y_b)$ and the end point is at $(x_e, y_e)$. This ...
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2answers
25 views

Polar Equation to Rectangular

$$r=\frac{9}{4 \cos θ − 3 \sin θ}$$ How do I do this? (Equation is in polar form.) I have already tried to do this, but I don't know how to finish it.
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1answer
36 views

Give a geometrical interpretation of the intersection of the planes with equations [closed]

Give a geometrical interpretation of the intersection of the planes with equations \begin{align} &x + y − 3 = 0\\ &y + z + 5 = 0\\ &x + z + 2 = 0 \end{align} what is a geometrical ...
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3answers
28 views

Through the point $A(4,5)$ a line is drawn.

Through the point $A(4,5)$ a line is drawn inclined at $45°$ with the $+ve$ X - axis. It meets $x+y=6$ at the point $B$. Find the equation of $AB$. My solution.. Equation of $AB$ ...
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vote
2answers
61 views

Curve equidistant to sine and cosine.

If I have the sine and cosine curves plotted, what would be the formula of the curve that is equidistant to both curves? Here's a picture of how it looks like. The original question comes from a ...
0
votes
2answers
69 views

Geometrical interpretation of solving a $3 \times 3$ system of equations

Solve the following system of equations and give a geometrical interpretation of the result. \begin{align*} x + y + z &= 6\\ 2x + y − 3z &= -5\\ 4x − 5y + z &= −3 \end{align*} I know that ...
2
votes
1answer
35 views

Plane $3x + y - z= 4$ touches the ellipsoid $2z^2 = \sqrt7(1 - 2x^2 -y^2)$

Show that the Plane $3x + y - z= 4$ touches the ellipsoid $2z^2 = \sqrt7(1 - 2x^2 -y^2)$ My attempt: First I tried to convert the equation of ellipsoid in general form and then further applying the ...
2
votes
1answer
78 views

condition for cones to be reciprocal

Question : Show that the cone $$ax^2 + by^2 + cz^2 - cxy - ayz - bzx = 0$$ is the reciprocal of the cone $$(a^2 - bc)x^2 + (b^2 - ac)y^2 + (c^2 - ab)z^2 - 2(a^2 + bc)yz - 2(b^2 + ac)zx - 2(c^2 + ab)xy ...
0
votes
1answer
30 views

Distance between two Polar-Coordinates

I choose two Points in Berlin with the coordinates: 1: lat: 52.511206 long: 13.546486 2: lat: 52.527501 long: 13.319206 With an online tool I got the ...
0
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1answer
30 views

Finding the Locus of Circumcentre

Let $P$ be a point on circumcircle of $\Delta ABC$, where $A=(3,4), B=(-3,4), C=(4,3)$. Let feet of perpendicular from $P$ to $AB$ and $AC$ be $Q$ and $R$, respectively. Then locus of circumcentre of ...
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2answers
44 views

How to find whether a point lies on a line which is in parametric form?

Does the point $(1,8,3)$ line on the line with parametric equation: $$x = 5 + 2t$$ $$y = 2 + 6t$$ $$z = 1 + 3t$$ I know how to solve if they give me a equation of a plane and ask whether ...
0
votes
1answer
13 views

A problem about affine spaces

Let A be an affine space,dim(A)=4. P,Q are planes from A. If dir(P)!=dir(Q),then P and Q are disjoint. Is this proposition true or false? I know that two planes are parallel if they are disjoint ...
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3answers
48 views

A question about an equation of a plane

Let $A=(1,3,1)$; $B=(1,1,1)$; $C=(2,0,1)$; $D=(1,-2,3)$. Determine the equation of a plane that passes through $D$ and is parallel with $(ABC)$. I know the fact that ...
2
votes
1answer
21 views

Find the radius of the circle with some given conditions.

A circle having centre at C is made to pass through the point $P(1,2)$ , touching the straight lines $7x - y = 5$ and $x + y +13 = 0$ at A and B respectively. Then find the radius of the circle. I ...
3
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1answer
40 views

Polar coordinates in taxicab geometry

We know that in euclidean $\mathbb{R}^2$ space polar coordinates are defined by $$r = \sqrt{x^2 + y^2}$$ $$\theta = \arctan\frac{y}{x}\text{.}$$ Now, geometrically we can think of it as of point, ...
2
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1answer
30 views

Finding the point on the ellipse under certain conditions

This is a kind of simple question, but it gives me hard time: An ellipse is given in coordinate system. It passes points $(a, 0)$, $(0, b)$, $(-a, 0)$, $(0, -b)$, where $a$ and $b$ are positive ...
0
votes
1answer
20 views

Rotating point by angle

Let $X = (c, 0)$. If I will rotate $X$ by, say, angle $\alpha = \frac{\pi}{4}$, how can I determine position of new angle? Will it just be $X' = (c + \cos\alpha, \sin\alpha)$?
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1answer
12 views

Degree measure of multiple polygons

I made this design on the Desmos calculator, and I was wondering what the quickest way was to find the degree measure of each individual angle. What I know so far: The measures of each of the ...
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votes
3answers
95 views

Area of extended triangle [closed]

I have three points $$(0,0),\ (1,1),\ (2,0)$$ and $k$, where $k$ is a number, in this task $k = 2$. I need to calculate the area of ​​the figure extending it points less than or equal to $k$. (In ...
0
votes
1answer
49 views

Calculate area of a figure extended from the unit square

I have four points $$(0,0),\ (0,1),\ (1,1),\ (1,0)$$ and $k$, where $k$ is a number, in this task $k = 1$. I need to calculate the area of ​​the figure extending it points less than or equal to ...
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0answers
20 views

Learn tracing of conics and concoids

A major portion of my course revolves around tracing of conics and concoids. But the explanation in my books is poor. I'm looking for some online notes/texts or videos to learn tracing of curves. I ...
0
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1answer
29 views

Tips to fix coordinates in analytic geometry.

I now know how useful analytic geometry can be in bashing geometry problems involving side lengths. Does anybody have any tips on how to fix coordinates to keep the solution from becoming too tedious? ...
1
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1answer
28 views

Graphing a regular pentagon

I just realized that I didn't know how to graph a regular pentagon with integer coordinates... What are some possible coordinates for a regular pentagon with the uppermost point at coordinate (0,0)?
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0answers
15 views

eccentricity of the conic

I'm given this question to find the eccentricity of this conic : $x^2 + ky = 0, k>0$ The given equation can be written as $x^2 = -ky$ now we can say compare this with the equation of parabola. But ...
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0answers
12 views

triangulation of a surface, adapted to curvature

This is about my printed models of mathematical objects. All of the designs that I've published so far consist of grids of bent ‘rods’, and in most of them the spacing of vertices depends on the ...
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2answers
24 views

A $2\times 2$ linear matrix transformation is conformal if and only if $c=-b ,d=a$ and $a,b \neq 0$

Let $T:\mathbb{R}^{2} \to \mathbb{R}^{2}$ a linear transformation represented by the matrix $A=\begin{pmatrix} a & c \\b&d \end{pmatrix}$ . Show that $T$ is conformal if and only if $c=-b ...
1
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1answer
44 views

Equilateral triangle with vertices whose coordinates on the Cartesian plane are integers. Does such a triangle exist? [duplicate]

Can you build an equilateral triangle on a Cartesian plane whose vertices only have integer values as their coordinates? Looking at the simplest example, i.e. a triangle with vertices (0,0), (1,0) ...
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2answers
86 views

Prove that the value of $(abc)-(ab+bc+ca)+3(a+b+c)$ is $0$

If the points $\big(\frac{a^3}{a-1}, \frac{a^2-3}{a-1}),(\frac{b^3}{b-1}, \frac{b^2-3}{b-1}) ,\big(\frac{c^3}{c-1}, \frac{c^2-3}{c-1}\big)$ are collinear for three distinct values of $a,b,c$ and ...
1
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1answer
32 views

parameterization of polar coordinates vectors

Assuming everything in 2D, if there is a circle with centre at origin, with radius R , we can write its cartesian equation as : x^2 + y^2 = R^2 . It's vector form in cartesian coordinates will be : ...
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0answers
20 views

In how many ways does a vector determina a hyperplane?

The perpendicular hyperplane is the *standard* way. The gradient of a differentiable function $f: \Bbb R^{n+1} \rightarrow \Bbb R$ gives another idea. Select one direction to be the height $h$. Then, ...
2
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3answers
36 views

If the reflection of the hyperbola $xy = 4$ in the line $x - y + 1 = 0$ is $xy = mx + ny + l$ find $m + n + l$

If the reflection of the hyperbola $xy = 4$ in the line $x - y + 1 = 0$ is $xy = mx + ny + l$ find $m + n + l$. I already solved it by taking a general point (more than one way possible for it) ...
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1answer
35 views

Point P such that perimeter is least

Given two points $A(-2,0)$ and $B(0,4)$ then find coordinate of point $P$ lying on the line $2x-3y=9$ so that perimeter of triangle $APB$ is least. Doing it by traditional calculus is making ...
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1answer
39 views

How to calculate triangle coordinates in cartesian plane?

My problem can be describe by following image: I know coordinates of an example P point. Say, they are equal to (8,8). I also ...
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0answers
25 views

prove that If and only if lines are perpendicular, the slopes are negative reciprocal.

I have to prove that If and only if lines are perpendicular, the slopes are negative reciprocal. I know to prove that if the lines are perpendicular,the slopes are negative reciprocal. But I dont ...
4
votes
3answers
39 views

If a family of straight lines is $\lambda^2 P+\lambda Q+R=0$ ,then the family of lines will be tangent to the curve $Q^2=4PR.$

I have read this theorem in my book but i do not know how to prove it. If a family of straight lines can be represented by an equation $\lambda^2 P+\lambda Q+R=0$ where $\lambda $ is a parameter and ...
3
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1answer
33 views

Focus of the Parabola

Find the Focus of $$(2x+y-1)^2=5(x-2y-3)$$. Clearly its a Parabola whose axis is $2x+y-1=0$ and since $x-2y-3=0$ is perpendicular to $2x+y-1=0$ Tangent at the vertex is $x-2y-3=0$.Also the Vertex is ...
1
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1answer
17 views

Would $f(x,y) = \frac{y}{x}$ give an actual graph of all the possible slopes of a function of 1 variable?

If you let $y$ stand for $\Delta y$ and $x$ stand for $\Delta x$. I wanted to plot this because I thought it would be interesting among other things to see the behavior of the function around $x=0$. ...
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2answers
39 views

what is the shortest distance between a parabola and the circle?

what is the shortest distance between the parabola and the circle? the equation of parabola is $$y^2=4ax$$ and the equation of circle is $$x^2+y^2-24y+81=0$$ if you can show graphically it will be ...
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vote
1answer
102 views

Shortest distance between two circles

What is the shortest distance, in units, between the circles $(x - 9)^2 + (y - 5)^2 = 6.25$ and $(x + 6)^2 + (y + 3)^2 = 49$? Express your answer as a decimal to the nearest tenth. So I know that ...
2
votes
2answers
30 views

the equation of two sides of a parallelogram are $2x-3y+7=0$ and $4x+y-21=0$ and one vertex is $(-1,-3)$. Find the other vertices.

First, I checked if the point $(-1,-3)$ is not a solution to the two given equations above so therefore none of those lines passes that point. Then, I solved for the lines parallel to equations above ...
0
votes
1answer
26 views

Line $mx + ny = 3$ is normal to the hyperbola $x^2 – y^2 = 1$

If the line $mx + ny = 3$ is normal to the hyperbola $x^2 – y^2 = 1$, then evaluate $\frac{1}{m^2}+\frac{1}{n^2}$. I compared given equation of normal to equation of normal at parametric point i.e ...
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2answers
25 views

The number of circles passing through the vertices of a triangle

I have read a book written by C.V Durell on Geometry. In this book I have found a lemma which states that there is one and only one circle that passes through three vertices of a triangle. I thought ...