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

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PARABOLA : Problem

Find the equation of line touching both the parabolas $$ x^2=-32y.......(1)$$ $$ y^2=4x.........(2) $$ i have equated slopes of both the parabolas and applied the condition that all the points on ...
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1answer
17 views

How are trigonometric ratios function of interior angles in a right angled triangle?

How can one assume that the ratio altitude/hypotenuse is a function of angle. For a general right-angled triangle--->Let: Hypotenuse$=c$ Altitude$=a$ Base$=b$ and angle opposite ...
3
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0answers
31 views

Find the radius of the circle for given conditions

A circle with center at origin passes through three points $P$, $Q$ and $R$ with the line segment $PQ$ as its diameter along $x$-axis. A line passes through $P$ intersects the chord $QR$ at point $D$. ...
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1answer
19 views

Find the gradient of lines joining the following pair of points.

If, $Gradient = \frac{(y_2-y_1)}{(x_2-x_1)}$ And, $(x_1,y_1),(x_2,y_2) = (p+3, p-3), (3p+4, p-5)$ Then, $(y_2,y_1) = ((p-5)-(p-3))$ $=((p-5)-p+3)$ $=(p-5-p+3)$ $=(-2)$ And, $(x_2,x_1) = ...
2
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0answers
43 views

Ring of germs of holomorphic functions at $0\in \mathbb{C}$

So I've been reading the book and they used a induction proof where they just state that for the base case the ring of germs of holomorphic functions on $\mathbb{C}$ is Noetherian. I looked at other ...
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1answer
18 views

Perturbation of tangent ball

As picture below, $A$ and $B$ are two balls, $\partial A\bigcap \partial B=\{k\}$, and $B$ contains $A$. How to show that $$ \forall h\in \partial B,\exists ~\varepsilon > 0 ~st~ A\subset ...
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1answer
27 views

if I know a point nearest the zero of a polynomial can I tell which zero it is? (finding intersect of $f(x)$ with a line)

I have a function $f(x)$ and two points $p_1$ and $p_2$. What I need to find is the point where $f(x)$ and the line defined by the two point intersect. I know what $f(x)$ is, $f(x) ...
3
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1answer
33 views

Question about the Jacobian of a function

Let $f:U\rightarrow V$ , $U$ and $V$ open subsets of $\mathbb{R}^2$, be a smooth function. Let $Jf_p$ be the jacobian of $f$ in the point $p\in U$ and set $M_p:=\sup\{|df_pv|:\|v\|=1\}$ and ...
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0answers
12 views

Requesting formulas about circum-circles

With the co-ordinates of the three vertices of a triangle given, we have nice looking formulas for the centroid and in-center. Do we have the same kind of formula(s) for the circum-circle or ...
2
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3answers
48 views

Why the Cartesian equations, are called “Cartesian”?

I've been studying analytic geometry and I'm wondering "Why the Cartesian equations, are called 'Cartesian,'" I know that the name is from the René Descartes philosopher. But in that one case why is ...
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1answer
16 views

Equation of a subspace given basis

Suppose we have a subspace expressed as the linear combination of two vectors (basis): $S = x * (3, -3, 1) + y * (5, 1, 3)$ How can I find the equation for the subspace (in this case, a plane ...
3
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1answer
72 views

A parallelogram between two points on a hexagonal lattice containing all the shortest paths

For any two points on a hexagonal grid with integer coordinates there is a unique parallelogram which contains all of the shortest paths (in terms of taxicab norm) between these points. See the ...
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1answer
20 views

Number of lattice points in triangle formed by x-axis, y-axis and given line

Given a line $ax+by=c$ where $a,b,c$ are positive integers. Is there any formula to find the number of points inside the triangle formed by this line, $x$-axis and $y$-axis? Points on the boundary ...
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2answers
34 views

Find closest Point to Another Point

How do I find the closest point to $(2,2)$ on the line CD, if C is $(3,2)$ and D is $(5, 3)$? How would I solve using linear algebra? Does it involve cross product and distance? Not sure how to solve ...
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0answers
17 views

Cohomology of $\mathcal{O}^*$ and projection map

Suppose $X$ is a complex manifold and $T$ a complex space (or complex manifold maybe) and let $\pi:T\times X \rightarrow T$ denote the projection. What are sufficient conditions on $X$ that make ...
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1answer
25 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 ...
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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 ...
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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
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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 ...
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1answer
16 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
52 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
25 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 ...
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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 ...
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2answers
71 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
22 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
35 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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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 ...
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2answers
57 views

Geometrical interpretation of solving a 3x3 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
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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
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1answer
62 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 ...
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1answer
27 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 ...
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1answer
26 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
42 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 ...
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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
47 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
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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
37 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
29 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 ...
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1answer
19 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
11 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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3answers
89 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 ...
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1answer
46 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
17 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
26 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
13 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
11 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 ...
0
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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
37 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) ...