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

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Which curve we will get under $\mathcal{A} \in M_2(\Bbb{R})$ from a unit circle

If I have a circle: $x^2+y^2=1$, It's parametric equation is : $$\begin{cases} x = \cos\theta \\ y = \sin\theta \end{cases}$$ under some transform: $A=\begin{pmatrix} a & b\\ c & d ...
3
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1answer
16 views

Compute direction of a cylinder by using 10 coefficients

I am wondering if anyone knows how to compute the direction of a cylinder by using the 10 coefficients. For example, we have the equation of a cylinder as ...
1
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0answers
27 views

Undergraduate research submission problem - Endorsement [on hold]

I have come up with a new approach to unit vector transformations. The new approach allows the reader to witness the process of transformation. Every step of the transformation process is accompanied ...
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1answer
13 views

Gaussian sums values

I have the following problem: Denoting $S(q,a,\chi ) = \sum_{x=1}^q \chi (x) e(ax/q)$, where $\chi $ is an arbitrary character modulo $q$, I have to prove $$\sum_{a=1}^q \vert S(q,a,\chi ) \vert ^2 = ...
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1answer
17 views

Deal with non standard form of conic

I want to know how can I calculate latus rectum, tangent at vertex, vertex and axes of a parabola whose equation is not standard. For example, the parabola: $$ 4x^2 - 4xy + y^2 - 10 y - 19 = 0 $$
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0answers
12 views

Cordinate geometry problem

The base of a triangle passes through fixed point $(1,1)$ and its sides are bisected at right angle by the lines $y^2 - 8xy - 9x^2 =0$. If the locus of its vertex is a circle of radius ${(k/2)}^{1/2} ...
3
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0answers
69 views

Calculate the flux of $\underline{v}$ across the boundary of the sector.

For $a\in(0,1)$, calculate without use of the divergence theorem the flux of $\underline{v}(x,y) = g(y/x)(-1/x,1/y)$ across the boundary of the sector $ S_a := \{(x,y)\in \Omega : 1\leqslant x^2+y^2 ...
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1answer
34 views

Find the equation of the parabola with its vertex on the line $2y-3x=0$?

Its axis of symmetry is parallel to the x-axis, and it passes through the two points $(3,5)$ and $(6,-1)$
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1answer
28 views

Paramaterizing a Parabola with $3$ points.

Let $A, B, C$ be vectors in $\mathbb R^2$. I want to show that the set $\{A+tB+t^2C\mid t\in\mathbb R\}$ defines a parabola in $\mathbb R^2$, but I'm having a hard time doing so, since I can't solve ...
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1answer
52 views

What is the equation of the reflections of a fixed point across all the tangents to a fixed circle?

Given a fixed circle "c" and a fixed point "A" (in the plane of the circle), draw the tangent to the circle at a variable point "X" (movable, but constrained to be on the circle), reflect "A" across ...
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0answers
33 views

What is the equation family of the projectile-motion-with-air-resistance eqn?

The general form of the equations of projectile motion with air resistance are (from here) $s_y(t) = -\frac{mg}{k}t + \frac{m}{k}(v_{yo} + \frac{mg}{k})(1 - e^{-\frac{k}{m}t})$ and $s_x(t) = ...
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0answers
19 views

Coordinates of the center of the sphere with given equation

I´m trying to solve this question, which asks for the coordinates of the center of the sphere $$ 4(x^2+(l-y)^2+z^2) = x^2+y^2+z^2 $$ I know the answer should be $(0, 4l/3,0)$. There's a picture of ...
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0answers
16 views

coordinate geometry problems

If $X \cos A +Y \sin A = p \,$ where $p = – \sin^2A/\cos^2A$ be a straight line Prove that perpendiculars on these straight lines from the points $(m^2, 2m)$ , $(Mm, M+m)$ , $(M^2, 2M)$ form a ...
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0answers
41 views

Asymmetric hyperbola-type curve? (for fitting to data)

I have this question: what would be the name and equation of a curve which resembles a parabola but has not the requirement of symmetry? So the general parabola equation is: $y=ax^2+bx+c$ I must ...
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2answers
122 views
+50

How to generate points uniformly distributed on the surface of an ellipsoid?

I am trying to find a way to generate random points uniformly distributed on the surface of an ellipsoid. If it was a sphere there is a neat way of doing it: Generate three $N(0,1)$ variables ...
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0answers
5 views

Homothetic center of two spheres

If the external homothetic center of two circles in the cartesian plane can be found with the following formula: $(x_e, y_e) = \frac{-r_2}{r_1 - r_2}(x_1, y_1) + \frac{r_1}{r_1 - r_2}(x_2, y_2)$ Am ...
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2answers
63 views

Equation of circle in terms of length of arc above $x$-axis

Say I have a circle centered at $(0,b)$ that passes through $(-5,0)$ and $(5,0)$ and has upper-half length $d.$ Now I've figured out that the equation of the circle is $$x^2 + (y-b)^2 = 5^2 + b^2$$ ...
0
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1answer
15 views

Parabola max. number

If the directrix and the tangent at vertex of a parabola are given then what is the maximum number of parabolas that can be drawn? Well according to me the answer should be 1 because the distance ...
4
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4answers
84 views

Equation of a line tangent to circumference

Discover the general equation of the tangent line to the circumference $x^2 + y^2 - 2x + 4y + 1 = 0$ by the point $(3,4)$. NO CALCULUS. by the circumference equation i discovered that $C(1, ...
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2answers
26 views

implicit equation for elliptical torus

I just wondering what the implicit equation would be if an ellipse with major axis a and minor axis b, rotating about the Z axis with a distance of $R_0$. The $R_0$>a and $R_0$>b which means the ...
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1answer
23 views

How to find the equation of conics given foci and directrices

Find the equations of the following conics, each with its centre at the origin. (a) A hyperbola with foci $(\pm4, 0)$ and directrices $x= \pm2$ (b) An ellipse with foci $(0, \pm4)$ and directrices ...
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3answers
61 views

Equation of line passing through point.

The straight line $3x + 4y + 5 = 0 $ and $4x - 3y - 10 = 0$ intersect at point $A$. Point $B$ on line $3x + 4y + 5 = 0 $ and point C on line $4x - 3y - 10 = 0$ are such that $d(A,B)=d(A,C)$. Find ...
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1answer
10 views

geometry of a hyperbola and circle drawn together

how to calculate the radius of a circle which is drawn below(inwards) the hyperbola curve touching it.need a relationship between these hyperbola and circle .If a circular object is place below the ...
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3answers
40 views

Find a vector minimizing the distance from set

Find a vector $\Pi_Z(x)$ minimizing the distance between $x=(5,10)\in\mathbb{R}^2$ and set $Z=\{(x,y)\in\mathbb{R}^2:x\ge0, y\le\sqrt{x}\}$
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0answers
67 views

Number of intersections of two circles

The equation of a circle C is x^2+y^2 -6x-8y-11=0. The number of real points at which the circle drawn with the points (1,8) and (0,0) at the ends of a diameter cuts the circle C is ? Results: ...
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2answers
34 views

Is there a function whose graph is contained in one quadrant of the coordinate plane?

Is there a function whose graph is contained in one quadrant of the coordinate plane? It should be related to maths and not physics. Please give me the equation and if possible its picture.
2
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1answer
31 views

How to find the equation for the line $t$, in the plane $\pi$ and concurrent to other 2 lines

The exercise says that $t$ is in the plane $\pi: x-y+z =0$ and is concurrent to the lines: $$r:\\x+y+2z=2\\x=y$$ and $$s:\\z=x+2\\y=0$$ I've transformed $r$ to the form: $$r:\\x = \lambda\\ y = ...
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1answer
15 views

Find the line $t$ that is concurrent to $r$ and $s$ and parallel to $MN$

I need to find the vector equation for the line $t$ that is concurrent to both: $$r:X = (1,1,-1)+\lambda(2,1,-1)$$ and $$s:\\x+y-3z = 1\\2x-y-2z=0$$ And also, $t$ is parallel to $MN$ when: $$M = ...
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1answer
26 views

Find the vector equation of the line parallel to the plane $\pi$, perpendicular to the line $AB$ and that intercepts $s$

I have the plane: $$\pi:2x-y+3z-1 = 0$$ $$A = (1,0,1), B = (0,1,2)$$ And $$s: X = (4,5,0) + \lambda (3,6,1)$$ I need to find a line that is perpendicular to $AB$, parallel to the plane $\pi$ and ...
0
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1answer
22 views

Find the equation of the plane that contains the line $r$ and makes an angle with $s$

I have the line: $$r:\\3z-x = 1\\y-1 = 1$$ And the plane makes an angle $\theta = \arccos \frac{2\sqrt{30}}{11}$ with the line: $$s:X = (1,1,0) + \lambda(3,1,1)$$ What I tried: From the equations ...
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1answer
23 views

Vectors - theory on cross product

If $X$ is a point on a line through $P$ and $Q$, $X=OX, P=OP, Q=OQ$ (all are vectors but $X$) then: $$X \times (Q-P) = P \times Q$$ I subbed in the $OX$, etc and simplified, but did not get each ...
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1answer
57 views

Why this isn't working? Find the points of the line $r$ that has the distance $\sqrt{\frac{14}{3}}$ from line $s$.

I have the line $$r:\\x+y=2\\x=y+z$$ and $$s:x=y=z+1$$ I need to find the points of $r$ that has distance $\sqrt{\frac{14}{3}}$ from $s$. What I tried: By using the formula for distance of a ...
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0answers
29 views

Find the symmetric equation of the tangent line to the curve

Find the symmetric equation of the tangent line to the curve defined by the intersection of $3x^2+2y^2+z^2=49$ and $x^2+y^2-z^2=10$ at the point $(3,-3,2)$. I know that when you obtain the partial ...
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2answers
192 views

How to find the number of squares formed by given lattice points?

Let us say that we are N integer coordinates (x, y) - what would our approach be if we were supposed to find the number of squares we could make from those given n points? Additionally, if we were to ...
0
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1answer
18 views

Finding circumcenter

In a triangle A(1,2) B(2,3) C(3,1) and $\angle A = cos^{-1}(4/5)$, $\angle B = \angle C = cos^{-1}(\frac{1}{\sqrt{10} }) $ Ordinate of circumcentre of the $\triangle$ is ? I have tried solving by ...
2
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1answer
47 views

Find the maximum possible area for the triangle

Two vertices of an isosceles triangle are (1,2) and (4,6). The inradius of the triangle is $\frac{3}{2}$. Find the maximum possible area for the triangle. My work, for the two possible structures of ...
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1answer
24 views

equation of major axis of an ellipsoid

What is the equation of 3 major axes of the following ellipsoid? $$ \begin{pmatrix}x & y & z\end{pmatrix} \begin{pmatrix} \alpha_1 & \beta_3 & \beta_2\\ \beta_3 & \alpha_2 & ...
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4answers
47 views

Find equation of a line perpendicular to the tangent of curve at a given point.

I need to find the equation to the line perpendicular to the tangent to the curve $y = x^3 -3x +1$, at the point $(2,3)$. Our teacher assigned us homework on stuff we haven't learned, so please if ...
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1answer
14 views

How does one obtain Hesse normal form of plane equation?

We have been studying the Hesse normal form of the plane equation, but the sketch of the plane in space given by the lecturer was horrible. Basically I ask you to explain me how does one obtain the ...
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0answers
26 views

What is the graph of the identity function in cartesian coordinates?

Why is that the graph of f(x) = x is the straight line that is the bisector of the first quadrant? (or, amounting to the same thing, the bisector of the third quadrant) By calculating the outputs for ...
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5answers
24 views

Find point on a line, given the line equation and distance from the origin [closed]

Given the line $y=3x+6$, how to find the coordinates of the points on the line which are $9$ units from the origin?
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1answer
36 views

What's the parametric equation for the general form of an ellipse rotated by any amount? Thanks.

What's the parametric equation for the general form of an ellipse rotated by any amount? Preferably, as a computer scientist, how can this equation be derived from the three variables: coordinate of ...
0
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0answers
30 views

Finding an ellipsoid equation

I want to find 3D equation of a falling droplet that I have considered it as an ellipsoid. I put two cameras, one in xy plane and another in zy plane to capture two projected views of the droplet and ...
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2answers
45 views

Distance of a Point from Hyperbola

Consider the part of hyperbola $H_{+}=\{(x,1/x)\colon x>0\}$ in the first quadrant, and $(a,b)$ any point in the plane (for sake of convenience, say $a,b>0$). If $(a,b)$ does not lie on the ...
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1answer
28 views

Locus of a point that satisfy a condition on the square of distances to two lines and their intersection

Find the locus of a point such that the square of its distance to the point of intersection of two perpendicular lines is equal to the sum of its distances to those lines. Assume $P(x,y)$ is any ...
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1answer
18 views

I want to know where I did wrong in finding the plane equation

I am asked to give 3 plane equation where the third plane will passes through the intersection of the first 2 planes and parallel to y axis. I came up with 2 plane equation which is also parallel to ...
2
votes
1answer
41 views

Geometric locus

The problem is: Let $A$, $B$ and $C$ be fixed points, and $α,β,γ$ and $κ$ are given constants, then the locus of a point $P$ that satisfies the equation $$α(AP)^2+β(BP)^2+γ(CP)^2=\kappa,$$ is a ...
0
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2answers
16 views

Describing vector equation geometrically

How would I describe geometrically the vector equation: $$\vec{x} = s(0,2,1) + t(1,1,-1) ,\qquad s,t \in \Bbb R$$
2
votes
2answers
36 views

find equations of an ellipsoid axes

I have an ellipsoid with the center point at the Origin and the following equation: $$\alpha_1 x^2+\alpha_2 y^2+\alpha_3 z^2+2\beta_1 zy+2\beta_2 xz+2\beta_3 xy=1$$ How can I find the equations of ...
0
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0answers
17 views

when you draw 1 altitude/ perpendicular bisector of an equilateral triangle, what can you form?

when you draw 1 altitude/ perpendicular bisector of an equilateral triangle, what can you form such that when you draw 4 equilateral triangles the foot of the perpendicular of the equilateral ...