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

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52 views

Center of Arc with Two Points, Radius, and Normal in 3D

I'm struggling to get the math to work out on this. I need to derive an alorithm for a program where I'm representing geometric entities. In this case, it's an arc. I would like to create the arc ...
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1answer
29 views

Finding the equation of locus

The co ordinates of any position of a moving point P are given by $$\left[\frac{(7t-2)}{(3t+2)} , \frac{(4t+5)}{(t-1)}\right]$$ where $t$ is a variable parameter.Find the equation of locus of $P$. ...
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1answer
39 views

Equation of a Pair of Straight Lines.

If $ax^2+2hxy+by^2$ be the two sides of the parallelogram and $px+qy=1$ is one diagonal then prove that the other diagonal is $y(bp-hq)=x(aq-hp)$. By reading the question I just understood that the ...
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3answers
35 views

If $PQ$ subtends right angle at the centre of ellipse then find $\frac{1}{OP^2}+\frac{1}{OQ^2}. $

$PQ$ is a variable chord of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ . If $PQ$ subtends right angle at the centre of ellipse then find $\frac{1}{OP^2}+\frac{1}{OQ^2}. $ Two points can be taken ...
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1answer
39 views

Locus of vertex of a rectangle

If from the vertex of a parabola $y^2 = 4ax$ a pair of chords be drawn at right angles to one another and with these chords as adjacent sides a rectangle be constructed , then we have to find the ...
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3answers
53 views

Finding the third vertex of an equilateral triangle

If two vertices of an equilateral triangle are (0,0) and (3,√3) then find the third vertex. The first thing I did was calculated the distance of the given points and tried to make an equation ...
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1answer
18 views

3 normals on a parabola

If $(x_1, y_1), (x_2, y_2)$ and $(x_3, y_3)$ be three points on the parabola $y^2 = 4ax$ and the normals at these points meet in a point then how will we prove that $$ \frac{x_1 -x_2}{y_3} + ...
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0answers
18 views

Radical axis/curve of two plane curves

Apart from circles which curves $C_1 =0 $ and $C_2 =0 $ have another common line /curve $ C_1=C_2 $ from which equal tangents can be drawn ? EDIT1: For example, an ellipse and parabola $( x - ...
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22 views

Example 1 Sec 13 in Munkres TOPOLOGY: A sphere contained in the intersection of spheres

Let $m, n \in \mathbb{N}$. Let $r_1, \ldots, r_m$ be given positive real numbers; for each $i=1, \ldots, m$, let $(x_{i1}, \ldots, x_{in})$ be $m$ given points in $\mathbb{R}^n$, and let $$S_i ...
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1answer
39 views

Geometric solution for the inverse kinematics problem (posted this in robotics but could not find the right answer as it is more mathematical )

I am working on an forward kinematics geometric solution. What it roughly means is that given a certain $(x,y,z)$ co ordinates the robot should compute the angles at which it should move its arms. I ...
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1answer
29 views

Finding minimal ratio of $AB/BO$

A line passes through the origin $O$ and cut the parabola $y=-\frac{x^2}{2}+1$ at point B in the first quarter, and also cut the line $y=-x+2$ at point $A$. Need to find points $A,B$ so that the ...
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4answers
55 views

How do I find the image of a point in a line in 3D space?

The question is, find the image of the point $(1, 6, 3)$ in the line $$\frac x1 = \frac {y-1}{2} = \frac {z-2}{3}$$ I want to know the general equation to find the image of a point in a line. ...
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2answers
51 views

Locus formed by point on a line intersecting 3 other lines in 3D

I got this particular question from an old test paper... Consider three lines given by $y-2=z+3=0$; $z-3=x+1=0$; $x-1=y+2=0$. Let $(\alpha,\beta,\gamma)$ be a point lying on a line intersecting ...
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0answers
25 views

Finding an angle inside a regular hexagon given a line that passes through two vertices

I am asked to find the angle $\alpha$ on this particular setup: The equation of the line is and point $A = (2,1,4)$. Is that really possible to find out? Maybe there is some data missing? ...
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1answer
37 views

Finding angle between straight lines whose direction cosines are implicitly given

Prove that the angle between the straight lines whose direction cosines are $l,m,n$ are given by $l+m+n=0$ and $fmn+gnl+hlm=0$ is $\pi\over 3$ if $1\over f$ +$1\over g$+$1\over h$=$0$. Also ...
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2answers
43 views

Feet per radian to feet per degrees?

How can we convert feet/radian to feet/degrees? I need to convert $-7600\sqrt 3 \frac{feet}{radians}$ I know the answer is: $-\frac{380 \pi} {9} \sqrt 3$ =$-230$ $\frac{feet}{degrees}$ approx But ...
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1answer
61 views

Find radius of the circle analytically

Given the circle as seen in the attached image, find the radius of the circle analytically. Is that even possible? I know it can be found numerically. If analytical solution does not exist, can you ...
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1answer
36 views

Parabola and straight line [duplicate]

If $m$ varies then find the range of $c$ for which the line $y=mx + c$ touches the parabola $y^2 = 8(x+2)$ . I tried Put the value $y = mx + c$ in the parabola equation and then done $\Delta = 0$ or ...
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3answers
126 views

Which particular pair of straight lines does this equation represent on putting $z=0$?

Suppose we have a joint equation of planes $8x^2-3y^2-10z^2+10xy+17yz+2xz=0$.Suppose we put $z=0$ we get a joint equation of pair of straight lines. Now which particular pair of straight lines does ...
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1answer
71 views

Bijection that preserve lines must be linear

There have been some past posts on this topic, but with no complete answer provided. Namely, if T is a bijection of the Euclidean plane that maps line segments to line segments (setwise) then T is an ...
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2answers
32 views

Area of Projection of Parallelogram

We are given a parallelogram $ABCD$ with $AB=\vec{u}$ and $AD=\vec{v}$. We know then that the area of $ABCD$ is given by $|\vec{u} \times \vec{v}|$. Show that the projection of $ABCD$ to a plane ...
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1answer
19 views

A question about classifying conics, from Garrity's book

For a general conic $$ ax^2 + bxy + cy^2 + dx + ey + h = 0$$ we see that, when rearranged and thought of as a quadratic in $x$, the discriminant of the resulting quadratic is $$\Delta_x(y) = (b^2 - ...
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1answer
53 views

General formula of Mobius transformation in the Cartesian plane.

I seem to be doing it wrong every time. time to learnbn the general formula for mobius transformations. But what is the general formula? suppose Given an Mobius transformation $$ f(Z) \ = \ ...
2
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1answer
25 views

Parabola with given points

Let if there is a parabola passing through some points eg $(0,1)$ , $(-1,3)$ , $(3,3)$ & $(2,1)$ Then if have we to find vertex and directrix . As there are two parrallel chords then the abcissa ...
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0answers
48 views

Construction new ellipse

Using a pencil the thread was pulled on the ellipse. Then the pencil started to rotate around the ellipse. How to prove that a new geometric figure which the pencil drew is also an ellipse (with the ...
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0answers
33 views

Triangle inequality in the Poincaré half plane

How can I prove the triangle inequality holds in the Poincaré Half Plane when given points $A$, $B$, $C$. Using the idea: $$d(A,B) + d(B,C) \ge d(A,C),$$ where $d(A,B)$ is the Poincaré distance ...
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4answers
64 views

Distance between $(-3, 0, 1)$ and the line $(2t, -t, -4t)$

Could you please help me with this problem? How do I calculate the distance between $(-3, 0, 1)$ and the line with the following parametric equation: $(x, y, z) = (2t, -t, -4t)$? I'm really lost ...
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3answers
43 views

How to find a straight line orthogonal to a curve?

We want to find the equation of line passing through $(3,6)$ and cutting the curve $y=\sqrt{x}$ orthogonally . I thought that it means we have to find a normal through that point equation of normal ...
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1answer
29 views

How to calculate the outward flux of a vector field through a cone?

Let $R$ be a region in the plane, and let $P$ be a point at a height $h$ above the plane. Form a cone by drawing lines from $P$ to each point on the boundary of $R$, and define a vector field by $x ...
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1answer
65 views

Parameterization of a torus

Given that the parameterization of a torus is given by: $x(\theta,\phi) = (R + r\cos(\theta))\cos(\phi)$ $y(\theta,\phi) = (R + r\cos(\theta))\sin(\phi)$ $z(\theta,\phi) = r\sin(\theta)$ and the ...
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0answers
15 views

symetrical coordinates of algebraic variety

Let $$M = \{ \mathbf{x} \in \mathbb{R}^n : x_i \geq 0 \}$$ and $c \in \mathbb{R}$, $\alpha_i \in \mathbb{Z} \setminus \{0\} $ for $i \in \{1,\dots,n\}$ $$ \mathcal{S} = \{ \mathbf{x} \in M : c = ...
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1answer
29 views

finding the equation of tangent lines to curves

Find the equation of the tangent to the curve $y = x^2 -6x +5$ at each point where the curve cuts the axis. Find also the coordinates of the point where these tangent line meet. I found the gradient ...
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1answer
12 views

finding equation of the normal

Find the equation of the normal to the curve $y= 4/x$ at point where $y=1/2$. Find the coordinates of the point where this normal cuts the x axis. I know that the curve cuts the x axis when $y=0$ ...
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1answer
36 views

Parabola conic section

Two tangents to the parabola $y^2= 8x$ meet the tangent at its vertex in the points $P$ and $Q$. If $|PQ| = 4$, prove that the locus of the point of the intersection of the two tangents is $y^2 = 8 ...
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1answer
25 views

Calculate the coordinates of the end point of a line that is inclined by a specified angle

I am trying to determine the the coordinates of the end point of a line. This line is inclined by $\theta$ . An example coordinates plane In this example we have $a = (4.5,4)$, $\theta = 45$ , $ab = ...
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1answer
23 views

Appolonius three circles problem, finding centre of tangent circle (analytic geometry)

Given 3 circles: $C_1$ centered at $(0,0)$ with radius 1 $C_2$ centered at $(a,0)$ with radius $a+1$ $C_3$ centered at $(-a,0)$ with radius $a+1$ (so $C_1$ is internaly tangent to both $C_2$ and ...
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0answers
14 views

Surface of a torus in terms of Legendre polynomials

The equation of a spheroid is $$\frac{x^2 + y^2}{a^2} + \frac{z^2}{b^2}$$ Its surface can be expressed as $$ r = a \left( 1 - \frac{2}{3} \epsilon P_2(\cos \theta) \right) $$ where $r$ is the ...
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1answer
41 views

Least value of $(α-β)$ if area is minimum.

The area of parallelogram formed by the lines $$x \cos \alpha +y\sin\alpha = p,$$ $$x \cos\alpha +y\sin\alpha = q,$$ $$x\cos\beta + y\sin\beta = r$$ and $$x\cos\beta + y\sin\beta = s$$ for given ...
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2answers
31 views

Calculus: Find equation that represents the set of all points that are equidistant from given three points (0,0,0) (2,4,3)(10,8,9)

So it is essential circumcenter problem in 3D that involves multivariable calculus. If you could at least help with the ideas or steps of tackling this problem that would be great PS: I made up the ...
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3answers
30 views

Circle of radius of Intersection of Plane and Sphere

The plane $x+2y-z=4$ cuts the sphere $x^2+y^2+z^2-x+z-2=0$ in a circle of radius? I tried putting value of y from plane in sphere but then I get a $zx$ term. How to proceed?
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1answer
46 views

What is the length of [BC]?

Let A , B and C be 3 points of a circle (c) P is the intersection of two tangents of the circle in points B and C Let (AB)//(CP) and AB=3 and BP=4 What is the length of BC Can someone give hint ! ...
3
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1answer
50 views

What is the Newton's general theory of diameters?

I was reading a book on Mathematics, which contained this topic. I was not able to grasp the concept. There was not much info on internet also. It was as follows: Let an $n$th order curve be ...
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3answers
43 views

Distance between point and line in point slope form on a plane

If I have an equation in point slope form $$y=mx$$ how can I use the perpendicular distance formula: $$\text{Perpendicular Distance} = \frac{\left | Ax_{1} + By_{1} + C\right |}{\sqrt{A^2 + B^2} }$$ ...
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2answers
50 views

Formula that describes the movement of a bishop in chess

I'm programming a chess game and I'm trying to validate the movements every player tries to make. Obviously, every piece can move differently and I've had no trouble validating their moves up until ...
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2answers
41 views

Polar radius of a general ellipsoid

Is there a proper parametrization of a general ellipsoid in spherical coordinates? The regular parametrization is this: $$x=a\cdot \cos\phi \cos\theta\\y=b\cdot \cos\phi \sin \theta\\z=c\cdot ...
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37 views

Hyperbola and 3 normals from point P

From any point P on the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ three normals other than that at P are drawn. Find the locus of the centroid of the triangle formed by feet of the normals. Do we ...
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1answer
53 views

Is $f(x+a) - f(a) = f(x) + f'(a) x$ an identity?

Given a differentiable function of $x$, denoted by $f(x)$; is $f(x+a) - f(a) = f(x) + f'(a) x$ an identity? For example, if $f(x)=x^2$, then it gives $(x+a)^2 - a^2 = x^2 + 2ax$, which is true. So, ...
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1answer
27 views

Image of a circle under conformal map $1/z$

The image of a circle under conformal map $1/z$ should be a circle, but how to prove it (or how to find the relationship between the two circles)? $z = x + iy = d + a\exp(i\theta)$, where $a$ is the ...
0
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1answer
16 views

Find the equations of Lines given Points and Angle

I have the following scenario. The coordinates of points B and D are $(10,0)$ and $(10,-10)$ respectively. I want to construct angle $\angle BFD = 45^\circ $. How can i find the coordinates of ...
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2answers
22 views

finding the coordinate given a distance with its coordinates

A point $P(x,y)$ has a distance $5\sqrt{2}$ units from $Q(4, -7)$ and a distance $\sqrt{106}$ units from $R(-6,5)$. Knowing that, find $P$. the image is exactly the set of problem that our professor ...