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

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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
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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
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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
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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 ...
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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
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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, ...
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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
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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
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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
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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? ${{e}^{f}}+{{e}^{g}}\equiv 1$
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0answers
39 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
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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
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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 ...
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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
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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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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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0answers
24 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 $F(x, y, z) = xyz$, ...
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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
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3answers
31 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 ...
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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) = ...
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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
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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
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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 * ...
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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
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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
131 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 ...
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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
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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
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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 ...
1
vote
1answer
55 views

Why 5 point determine a conic?

How to prove that any five points, of which no 3 are colinear, there is a single conic that passes through all of them ? (I have to start out with the equation $$Ax^2+Bxy+Cy^2+Dx+Ey+F=0$$ but i don't ...
0
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0answers
30 views

Finding total number of lattice points on a circle and one of the coordinate values

I was wondering if it was possible to quickly find the total amount of lattice points on a circle, given its equation (origin and radius), and I was also a bit confused on how to find a particular ...
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0answers
87 views

External Division Of Line Segment Or Negative Internal Division

I'm sort of confused with external division of a line segment. When someone says that P divides a line AB in ratio say $2:1$ externally then it means distance of A from P is twice P from B. But in ...
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1answer
29 views

How do I confirm my answer for the orthocenter of a triangle?

I know how to find the orthocenter, but how do do I check to see if I have the right answer? For instance, once I find the centroid of a triangle I can use the centroid formula(a+b+c)/3 to check my ...
1
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0answers
36 views

'Smooth' p-adic analysis (perhaps via toposes)

There are sensible theories of analytic functions on non-Archimedean fields (rigid analytic spaces, Berkovich spaces), but these are modeled after complex analysis. I'm curious to what extent there ...
1
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2answers
21 views

Question on Straight Lines invoving Family of Lines and Angle Bisectors.

Lines $L_1$ ($a_1x+b_1y+c_1$) and $L_2$ ($a_2x+b_2y+c_2$) intersect at a point $P$ and subtend an angle $\theta$ at $P$. Another line $L$ makes the same angle $\theta$ with $L_1$ at $P$. Find ...
0
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2answers
19 views

Writing domain set for function F

For this function, I plot it. And I show the areas where $F$ take positive and negative values on the graph. Well, how can I write the set which shows the areas that F be positive and F be ...
3
votes
3answers
40 views

Shortest distance between point and surface

I wonder if there exists a good enough formula to compute the shortest distance between a point $P=(x_0,y_0,z_0)$ and a surface $\pi$ defined by $F(x,y,z)=0$. There is a lot of simmilar questions in ...
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2answers
31 views

showing global min point

let's $F(x)= x'Ax + a'x $ where $x'Ax$ is a quadratic form and $a'$ is defined as a vector. $$A:= \left[ \begin{matrix} 6 &1&1 \\ 1&2&0 \\ 1&0&4\end{matrix}\right] $$ Does ...
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1answer
51 views

some functions, ploting them and defining positive and negative areas

I have four following functions. I need to plot them and define positive and negative defined area. I have done most of them. But, i want you to check them, and correct them if false. (1) ...
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1answer
59 views

Inequations about curvature tensor.

How to get two inequations 1 and 2? At the 1, I don't know how to use the Cauchy inequation . At the 2, I absolutely start from where.. ...
2
votes
1answer
39 views

Coordinates of two missing vertices of a sqaure

I'm given the coordinates of two vertices of a square: $(5,1)$ and $(8,5)$ and I'm asked to find the other two. I suppose there are two squares which meet these conditions. I can obtain a point ...
0
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1answer
46 views

What does one mean by magnitude of normal in co-ordinate geometry?

Basically whenever I imagine a surface, by normal at a point, I mean a straight line perpendicular to the surface at that point which has an infinite length as straight lines do have. But how does ...
2
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2answers
83 views

Circles tangent to a parabola

For the past two weeks I was struggling with solving the following problem. Description of variables: $(x_n,y_n)$ - center point of the circle $C_n$ $r_n$ - radius of the circle $C_n$ Given the ...
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1answer
28 views

Suppose there exist exactly $n$ circles with non-zero radius in the plane tangent to all the three lines,then the possible values of $n$ is/are

Three distinct lines are drawn in a plane.Suppose there exist exactly $n$ circles with non-zero radius in the plane tangent to all the three lines,then the possible values of $n$ is/are ...
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2answers
51 views

Do two circles always have a radical axis?

Do two circles always have a radical axis? I came across this question in my book.I think every pair of circles have a radical axis,but my book answer says NO,not every pair of circles have a ...
1
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3answers
52 views

A plane,sphere and line all in an inequality

$x+y+z=4$ $x^2 +y^2+z^2=6$ If $x$,$y$ and $z$ are real numbers that satisfy the equation above, and that they lie in the range of $[a,b]$,then find values of both a and b. My attempt:- Instead of ...