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

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2
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1answer
22 views

Show that there is a common line of intersection of the three given planes.

Let $x-y\sin\alpha-z\sin\beta=0,x\sin\alpha-y+z\sin\gamma=0$ and $x\sin\beta+y\sin\gamma-z=0$ be the equations of the planes such that $\alpha+\beta+\gamma=\frac{\pi}{2}$,(where ...
0
votes
1answer
37 views

The new position of $O,$ when triangle is rotated about side $AB$ by $90^\circ$ can be

Consider the triangle $AOB$ in the $xy$-plane where $A\equiv(1,0,0);B\equiv(0,2,0);$ and $O(0,0,0)$.The new position of $O,$ when triangle is rotated about side $AB$ by $90^\circ$ can be ...
1
vote
1answer
42 views

Find the equation of the image of the plane $x-2y+2z-3=0$ in the plane $x+y+z-1=0$.

Find the equation of the image of the plane $x-2y+2z-3=0$ in the plane $x+y+z-1=0$. I have no idea how to find the image of a plane in another plane. Please help me.
0
votes
2answers
35 views

Find the leg of an altitude in a triangle

The vertices of $ABC$ are $A(8,5)$, $B(0,1)$ and $C(9, -2)$. Find the point where the altitude from $A$ intersects $BC$. Progress: I have found the equation of the altitude from A to BC, and that ...
0
votes
1answer
17 views

A parallelopiped is formed by planes drawn through the points $(1,2,3)$ and $(9,8,5)$ parallel to the coordinate planes

A parallelopiped is formed by planes drawn through the points $(1,2,3)$ and $(9,8,5)$ parallel to the coordinate planes then which of the following is not the length of an edge of this rectangular ...
0
votes
1answer
21 views

A rod of length $2$ units whose one end is $(1,0,-1)$ and the other end touches the plane $x-2y+2z+4=0$

A rod of length $2$ units whose one end is $(1,0,-1)$ and the other end touches the plane $x-2y+2z+4=0,$ then find the center of the region which the rod traces on the plane. The rod sweeps out the ...
0
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0answers
52 views

Find the equation of the plane which bisects that angle between the given planes which is acute and which contains the origin.

Consider the planes $3x-6y+2z+5=0$ and $4x-12y+3z=3$.Find the equation of the plane which bisects that angle between the given planes which is acute and which contains the origin. The two bisectors ...
2
votes
2answers
69 views

Finding arc length parametrization of a parabola

Suppose we have a parabola of equation $y = x^2$ in a given Cartesian coordinate system. An obvious parameterization of it is the system $x = t$, $y = t^2$, but there are infinite other possibilities, ...
0
votes
0answers
25 views

Analytic geometry question - finding an equation of a line in 3D

I need to find the equation of the line that intersects perpendicularly the line: $$ \frac{x+1}{2}=-y=\frac{z-2}{3} $$ and passes through : $(2,3,1)$. So I know that the line should be of the form: ...
1
vote
1answer
19 views

two-part partition of the unit Euclidean closed ball

In a Greek maths forum I found the following problem: problem: Let $\mathcal{P}=\{A,B \}$ be a partition of the $n$-dimensional Euclidean closed ball ($n>1$), prove that at least one ...
0
votes
1answer
19 views

Finding three new vectors pointing towards the vertices of a regular tetrahedron, with one vector given.

Explanation of the problem: I have a MATLAB program, which produces two vectors in a 3D coordinate system. The origin for both vectors is (0,0,0). The vector's endpoints are located on the unit ...
0
votes
1answer
93 views

Equations of the two lines through the origin which intersect the line $\frac{x-3}{2}=\frac{y-3}{1}=\frac{z}{1}$ at an angle of $\frac{\pi}{3}$

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 an angle of $\frac{\pi}{3}$. Let the direction ratios of the two ...
3
votes
4answers
105 views

Equation of a Straight Line sum

Okay, here is the question: A straight line makes on the coordinate axes positive intercepts whose sum is 7. If the line passes through the point (-3,8), find it's equation. I spent an hour in ...
0
votes
2answers
30 views

The equation of the plane which passes through the point of intersection of two space lines and at greatest distance from the point $(0,0,0)$

The equation of the plane which passes through the point of intersection of lines $\frac{x-1}{3}=\frac{y-2}{1}=\frac{z-3}{2}$ and $\frac{x-3}{1}=\frac{y-1}{2}=\frac{z-2}{3}$ and at greatest distance ...
3
votes
1answer
57 views

What do $l+p$ and $lp$, where $p$ is a point and $l$ is a line, mean in geometery?

I am looking at a graph theory problem that describes the partite sets of a bipartite as two copies of the $(m+1)$-dimensional vector space over the finite field $\mathbb{F}_{p^n}$ ($p$ is prime and ...
1
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1answer
68 views

Area of the triangle determined by the line $x+y=3$ and the bisector of angle between the lines $x^2-y^2+2y=1$

What is the area of the triangle formed by the lines $x+y=3$ and angle bisectors of pair of straight lines $x^2-y^2+2y=1$ . I found the intersection point of these equations $(1,2)$ but not getting ...
1
vote
1answer
31 views

Foot of perpendicular on a chord of a conic

For a standard ellipse, a chord subtends an angle of $90^{\circ}$ with the centre $(0,0)$ . To find the locus of the foot of perpendicular to this chord from the centre of the ellipse, I wrote the ...
-2
votes
2answers
49 views

Find the center and radius of the circle whose equation is $x^ 2 + y^ 2 - 6x - 2y + 4 = 0$ [closed]

Find the center and radius of the circle whose equation is $$x^2 + y^2 - 6x - 2y + 4 = 0$$
2
votes
1answer
33 views

Ellipse and two tangent lines

Given ellipse(${x^2 \over a^2}+{y^2 \over b^2}=1$) and a point $(x_0,y_0)$ We draw two tangent lines to the ellipse that are going through $(x_0,y_0)$ Find the equation of the straight line connecting ...
0
votes
1answer
41 views

If $P_1,P_2,P_3$ lie on the circle $x^2+y^2=1$,then prove that $P_4$ lies on the circle.

Given $4$ points $P_1,P_2,P_3,P_4$ on the coordinate plane with origin $O$ which satisfy the condition $\vec{OP_1}+\vec{OP_3}=\frac{3}{2}\vec{OP_2}$ and $\vec{OP_2}+\vec{OP_4}=\frac{3}{2}\vec{OP_3}$ ...
1
vote
2answers
22 views

a property for points in convex hull

Let $A\subset\mathbb{R}^2$ and $b=(b_1,b_2)$ is in the convex hull of $A$. Prove that for any $x=(x_1,x_2)\in\mathbb{R}^2$, there exists $a=(a_1,a_2)\in A$ such that $a_1x_1+a_2x_2\le b_1x_1+b_2x_2$. ...
3
votes
2answers
56 views

Find specific 4 curves touching $y=\cos10x+\cos21x$.

The following is the graph of $y=\cos10x+\cos21x$. You can see that there seems to be four curves that can touch this graph. I tried $y=\cos(x/2+\pi/2\pm\pi)+1$ and $y=-\cos(x/2\pm\pi/2)-1$: But ...
0
votes
1answer
21 views

How to find the equation of the plane that tangent to this surface?

Find the the equation of the plane that tangent to $x^2+2y^2+4z^2+xy+3yz=1$ and is paralel to $y=0$ plane first I found the gradient vector $\nabla f\left( x,y,z\right)=(2x+y)i+(4y+x+3z)j+(8z+3y)k $ ...
1
vote
3answers
49 views

Find the equations of the line of intersection of the following planes

Find the equations of the line of intersection of the following planes $2x − 3y + 2z = 5$ and $x + 2y − z = 4$. So i first put this in the normal vector form $\langle 2, -3, 2\rangle$ $\langle 1, ...
-1
votes
1answer
27 views

Algebraic coordinate geometry sl lonely question no 27 example 1.

please help solving these questions please if there is an issue with question then please comment below I am new to use this site. I don't even know how to solve the question 27) prove that a ...
0
votes
1answer
28 views

A point $P$ inside the tetrahedron is at the same distance $r$ from the four plane faces of the tetrahedron.Find the value of $r.$

The position vectors of the four angular points of a tetrahedron $OABC$ are $(0,0,0);(0,0,2);(0,4,0)$ and $(6,0,0)$ respectively.A point $P$ inside the tetrahedron is at the same distance $r$ from the ...
1
vote
1answer
44 views

Equation of 3 circles touching each other is given, what is the equation of a circle touching all 3?

Equation of 3 circles touching each other is given, what is the equation of a circle touching all other 3? does it matter that there are 2 circles that can touch all other 3 circles, one being ...
2
votes
3answers
33 views

Equation of locus

Point P$(x, y)$ moves in such a way that its distance from the point $(3, 5)$ is proportional to its distance from the point $(-2, 4)$. Find the locus of P if the origin is a point on the locus. ...
0
votes
2answers
31 views

Finding a point that is a certain distance away from a segment

I have two endpoints $(x_1, y_1)$ and $(x_2,y_2)$ of a line segment. I want to extend the existing segment by a length of $d$ on just one side of the segment. What are the coordinates of the new ...
0
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0answers
16 views

Is there such kind of entires and non constants functions? [duplicate]

Which characteristic of entire functions allows us to have the following expression? ${{e}^{f}}+{{e}^{g}}\equiv 1$
0
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0answers
40 views

Find the position difference between two transformations of a line

I am trying to find the new position of the points in a line, after certain changes. The line is simple: coordinate locale (0,0) to (len, 0); it has a position in the scene, and a rotation. If ...
2
votes
1answer
65 views

An inequality related to lattice points 'around' a circle

Take a circle of radius $r$ with centre at the origin such that $r^2=N_1^2+N_2^2$ for $N_1,N_2\in\mathbb{N}$. Consider a lattice coordinate $(a,b)$ such that $a\in(-r,-2)$ and define $b$ to be the ...
0
votes
1answer
23 views

problem related to the slope of a line.

What is the slope of the line given by $\sqrt{x^2+4y^2-4xy+4} + x-2y=1$ . Not getting any start . Only observed we have $(x-2y)^2$ under the root . NOTE: root gets over after 4 so please dont ...
1
vote
2answers
59 views

Given locus is a circle, prove two lines are perpendicular

Let $l_1$ and $l_2$ be two lines in the plane. The locus of all points $P$, such that the sum of squares of the distances of $P$ to $l_1$ and $l_2$ is constant, is a circle. Prove that $l_1$ and $l_2$ ...
3
votes
2answers
33 views

Prove that points A, B, K and L lie on a circle $c$

In an acute-angled triangle ABC with height CD, K and L are orthogonal projections through D respectively on AC and BC. Prove that points A, B, K and L lie on a circle $c$. I tried to prove that ...
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votes
2answers
21 views

Find the projection of the point on the line

Solve the equation of the projection of the point $A(1,2,8)$ on the straight line $p$ with the property: $$p=\frac{x-1}{2}=\frac{y}{-1}=\frac{z}{1}.$$
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vote
0answers
27 views

Finding the maximum volume of a box bounded by a plane?

The box is in the first octant, and one corner is located at the origin while the opposite corner is located on the plane $x+2y+3z=6$. My approach was to write the volume as $F(x, y, z) = xyz$, ...
0
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2answers
35 views

Point addition not Allowed

In what Structure point addition is not allowed and that makes points different from vectors.I mean in any Field or even Group i can add without problem but i have seen people saying point addition ...
0
votes
3answers
32 views

Why is the maximum increase given by $||\nabla f(x, y)||$?

I understand the steps of the proof in the book, but I don't see intuitively the of maximum increase at a point $P$ must be given by the $||\nabla f(x, y)||$. A graph has infinite directional ...
2
votes
1answer
35 views

Fastest computation to find out if two vectors intersect (programming problem)

I'm trying to write a program that should solve a 12x12 rush hour problem: I won't go in the details of this program to much. The program already works for 6x6 puzzles, but for 12x12 puzzles, it is ...
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votes
2answers
18 views

How can I find $\theta$ when converting an equation to cylindrical coordinates?

The equation is $x^2+y^2=4y$ and I need to convert it to cylindrical coordinates Here is what I did: $x = 2r\cos(\theta)$ $y = 2r\sin(\theta)$ $2r^2\cos^2(\theta)+2r^2\sin^2(\theta) = ...
1
vote
2answers
52 views

Reflection of $x=1$ about $x+y=1$

A ray of light travels along the line $x=1$ and gets reflected by a mirror on $x+y=1$. Find the equation of the reflected ray. $$$$ I am to solve this problem using only ...
1
vote
1answer
26 views

Point of intersection between two circles how do I get the point?

Circle1 with $(1,1)$ and $r=1$ Circle2 with $(3,2.5)$ and $ r=2$ Best way to calculate the intersection without a calculator on a piece of paper, I tried many ways which I saw on the internet and ...
0
votes
1answer
30 views

Find the coordinats of a triangle after rotation [closed]

How to calculate new coordinate of a 2d triangle rotated by Q degrees? We confused that x = old X - center of mass X y = old Y - center of mass Y x = x * cos(Q) - y * sin(Q) y = x * sin(Q) + y * ...
1
vote
1answer
32 views

If two vectors are normal to the same plane in $\mathbb{R}^3$, must they then be parallel to each other?

Following this article on MathWorld define the plane passing through a point $x_0$ perpendicular to a vector $n$ as the set of all points $x$ satisfying $$n \cdot (x - x_0) = 0.$$ Define a normal ...
0
votes
1answer
47 views

Analytical Geometry Tangents To Circle

I was solving one question coordinate geometry when i encountered this. I had to find slopes of tangents from a point to a circle. I applied condition of tangency that any line y=mx + c is tangent to ...
7
votes
1answer
132 views

Area of triangle in determinant form

Area of triangle with vertex $(x_1,y_1),(x_2,y_2),(x_3,y_3)$ is given by : $$\frac{1}{2}\begin{vmatrix} x_1 & y_1 & 1\\x_2 & y_2 & 1\\x_3 & y_3 & 1 \end{vmatrix}$$ In this ...
0
votes
0answers
12 views

Visualizing the coefficients of barycentric coordinates

I am poor in mathematics and please help me visualize the geometric interpretation of formula in the code, The problem is there are three points in a 2d plane ...
2
votes
2answers
134 views

Find the highest and lowest points on the ellipse of intersection of the cylinder $x^2+y^2 = 1$ and the plane $x+y+z=1$

Find the highest and lowest points on the ellipse of intersection of the cylinder $x^2+y^2 = 1$ and the plain $x+y+z=1$ Hi i was doing this question but i'm not sure i was right. Does this make ...
1
vote
1answer
34 views

Algebraic solutions for Poincaré Disk arcs

Given two points on the Poincaré Disk, there is a single straight line or arc that passes through them and that is orthogonal to the unit circle. Using compass and straightedge methods, one can easily ...