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

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2answers
30 views

Shortest distance form a point to a plane [on hold]

Let $T$ be the plane $−2x+y+z = 14$. Find the shortest distance d from the point $P_0=(−2, 3, −1)$ to $T$, and the point $Q$ in $T$ that is closest to $P_0$.
1
vote
1answer
19 views

need explanation of what exactly is a directrix & focus?

((I'm not asking why do we need to know conic sections etc.) Like other similar questions.) I actually love math & currently learning conic sections in class, neither my textbook or teacher ...
2
votes
2answers
30 views

Area of triangle inscribed in a parabola

How can u prove that the area of the triangle inscribed in a parabola is twice the area of the triangle formed by the tangents at the vertices?
0
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1answer
22 views

Centroid of triangle formed by co-normal points

How can you prove that he centroid of a triangle formed by 3 co-normal points lies on the axis of the parabola?
2
votes
2answers
37 views

Orthogonal tangents to an ellipse

This is the problem I found back in the first year in the university. Suppose we have a non-degenerate (i.e. not a point and not an empty set) ellipse $E\subset \Bbb R^2$. Now define a set $D$ by a ...
2
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3answers
59 views

The maximum from a point outside an ellipse to a ellipse.

In the $xOy$ axes, Assume there is an ellipse $$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$$, and a point $A(0,t)$ ($t$ is a constant )outside the ellipse. Assume $P$ is a point in the ellipse. Find the ...
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4answers
45 views

Equal perimeter and area

Find all triangles of which perimeter and area are numerically equal. I have got solution for right angle triangles but not of others
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1answer
48 views

How find this this distance $d_{1}d_{2}=b^2$

On the plane we have two points $A(\sqrt{a^2-b^2},0),B(-\sqrt{a^2-b^2},0)$ with $a>b>0$ and the line $L$, of which the equation is given ...
1
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1answer
25 views

Is there another way to solve the value field of a parameter of an line.

Assume $P$ is a point in line $x+y=m$, where $m \in \Bbb{R}$. There are two points $A,B$ in circle $$x^2+y^2 = 10$$ such that $PA$ and $PB$ are tangent lines of the above circle. If line: $x+y=m$ has ...
0
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0answers
17 views

Finding the equation of a plane by using point-to-point distances

Assume that we have a plane $P_1$ whose equation is known. I need to find the equation of plane $P_2$. If we choose a point set $N = \{n_1, n_2, ...\}$ on $P_1$ and another point set $M = \{m_1, m_2, ...
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2answers
30 views

How to compute point from {length and angle}

How to compute point from {length and angle}?
0
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1answer
37 views

The relation between the radiuses…

Find $\frac{R}{r}$ where $R$ is the radius of the circumscribed circle of a trapezoid and $r$ is the radius of the inscribed circle of this trapezoid. Thank you!
1
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1answer
54 views

Problem concerning inscribed and circumscribed circles…

Can you please help me solve this really difficult problem: Find R/r where R is the radius of the circumscribed circle of a trapezoid and r is the radius of the inscribed circle of this trapezoid. ...
1
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1answer
31 views

Perpendicular form of the straight line equation.

There are 5 to 6 standard forms of the straight line equation. for example slope intercept form, two intercept form, point slope form and perpendicular form. I have clear visualization of all forms ...
0
votes
1answer
19 views

Lattice Points on a straight line.

To find: The number of lattice points in the 1st quadrant, lying on straight line: 3x 5y = 283. -I tried this question a lot many times. The long substitution method becomes tedious. Can u please ...
0
votes
2answers
26 views

sides of two triangles which have different areas

consider 2 triangles like $\bigtriangleup ABC \quad and \quad \bigtriangleup \acute{A}\acute{B}\acute{C}$, which $S_{\bigtriangleup \acute{A}\acute{B}\acute{C}} \leq S_{\bigtriangleup ABC}$.(S stands ...
0
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2answers
51 views

How to show that a regular pentagon can't have all coordinates rational

This is pretty straightforward if we're allowed to use trigonometry, so I guess my question is Are there any nice (trigonometry-less) proofs of the fact that a regular pentagon in the plane must ...
0
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2answers
27 views

Find perpendicular vectors in subspace of $V_{3}$

Find all vectors of $V_{3}$ which are perpendicular to the vector $(7,0,-7)$ and belong to the subspace $L((0,-1,4), (6,-3,0)$. As a note, this is an extra question of a long exercise, the ...
0
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2answers
36 views

Prove $\sin^2 A = \sin^2 B \sin^2 C - 2\sin B \sin C \cos A$

I am asking for help with this proof: Given $\triangle ABC$. Prove that $\sin^2 A = \sin^2 B+ \sin^2 C - 2\sin B \sin C \cos A$
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2answers
55 views

$\overline{x} \times \overline{a} = \overline{b}$ has a solution when $ \langle\overline{a},\overline{b} \rangle =0$

I'm trying to solve this exercise: Let $\overline{a} \neq \overline{0}$, $\overline{b}$ be two vectors of the Euclidean vector space $V_{3}$. Prove the equation $\overline{x} \times \overline{a} = ...
0
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1answer
13 views

How can I find the coordinates of a point which is the reflection of a point about a line in 3D

I am currently working on a project on Matlab and I need to find the coordinates of a point which is reflected about a line. I know how to do it in 2D but in 3D things are getting ugly. So, we have a ...
3
votes
2answers
27 views

Scalar times Point + Scalar times Point?

Let $P$, $Q$ be a pair of points in the Euclidean plane and let $t_1$, $t_2$ be a pair of scalars. My textbook says that the following operations are nonsense: $$P + Q\\ t_1 \cdot P$$ However $t_1 ...
0
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1answer
29 views

How to find the equation of a parabola with vertex on the line y = -3x?

Its axis are parallel to the y-axis and passing through (-7,13) and (5,1).
0
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2answers
51 views

Find the tangents to the following curve from the given point.

2x^2 + y^2 = 54 from (10,1) P.S. I still don't study calculus. This lesson is from analytic geometry and I have no idea how to solve it because my professor didn't teach it. So if someone could tell ...
1
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1answer
28 views

A detailed basic-level explanation of equations of lines and planes in 3-d geometry

I've searched multiple blogs but couldn't find anything helpful for my level. (I'm in the 12th grade learning about vectors in maths). I basically need some thorough explanations regarding plane and ...
0
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1answer
18 views

points in the 3-dimensional space

Let $A=(a,b,c)$, $B=(d,e,f)$ and $C=(g,h,i)$ be points in the $3$-dimensional real vector space. It is well known that we can consider a new referential where we can see these points as ...
6
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0answers
62 views

Knight's metric: ellipse and parabola.

Knight's metric is a metric on $\mathbb{Z}^2$ as the minimum number of moves a chess knight would take to travel from $x$ to $y\in\mathbb{Z}^2$. What does a parabola (or an ellipse) became with this ...
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0answers
29 views

Three dimensional geometry problem

Find the distance of the point $P$ $(0,2,3)$ from the line $\frac{x+3}{3} = \frac{y-1}{2} = \frac{z+4}{3}$ using the formula $$((x_1-\alpha)^2 + (y_1-\beta)^2 + (z_1-\gamma)^2) - ((x_1-\alpha)l + ...
0
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0answers
9 views

What's the point in the general form of a plane equation?

So I've been reading up on my maths (Mathematics for Computer Graphics by John Vince FYI) and come to analytic geometry and I have a question. Why define the Cartesian form of a plane equation as: $$ ...
1
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1answer
31 views

exercise with lines algebra

This is an exercise I really can't solve by myself. 1) Let A(1,-1,0) a point on a line (e) line 2) Let (d) be a line perdicular to (e), given by a parametric equation. How I can find the equation ...
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0answers
22 views

points of intersection of two circles drawn on a sphere

How does one write and thereafter solve the equations to find the points of intersection of two overlapping circles drawn on the surface of a sphere? I am looking for a simple understandable solution ...
2
votes
2answers
65 views

check if point is on a plane (using Heron formula ?)

Is this true that if any of parameters a, b, c, d is equal to sum of three others then 4 points are on same plane? I am given 4 points in 3 dimensional space. Is this correct to state that all 4 ...
1
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2answers
21 views

Three coplanar vectors - proving a statement

Is it true that three vectors $\vec{u}, \vec{v} , \vec{w} $ lie on the same plane if and only if there exists constants $A,B,C,D$ for which $A\vec{u} + B\vec{v} + C\vec{w} +D =\vec{0} $ ? If so, how ...
0
votes
1answer
33 views

Analytic geometry and calculus question

The tangent to the curve $ x^{0.5} + y^{0.5} = a^{0.5}$ ,$(a>0)$ intersects the $x$ axis in point $M_1$ , and the $y$ axis in point $M_2$. Prove that for any point on the curve $|OM_1|+|OM_2| = ...
1
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2answers
24 views

Analytic geometry and calculus combined question

Show that the equation for the tangent with the slope $m$, $(m≠0)$ to the parabola $y^2 = 4px$ is $y = mx + \frac{p}{m}$. How this is done? What is the method for proves of this kind?
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2answers
36 views

Finding root for the segment - found the formula but it doesn't work for some values - wrong formula?

I have the segment, defined as $(x_1, y_1)$, $(x_2, y_2)$. I know that $y_1\ge 0$ and $y_2 < 0$. I want to compute the root point for that segment. I decided to do it that way: ...
1
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1answer
50 views

How to find out if four points are on the same plane, only by using distances?

There is a method called Cayley-Menger determinant in order to find if 3 points are collinear, 4 points are coplanar etc. provided that all the pairwise distances are given. However, in 2-D, there is ...
2
votes
3answers
38 views

Analytic geometry and calculus mixed question

Find the normal equation to the graph $ y = 2x^2$, that goes through the point $ (1,0.25)$. How this is done? Im not even really sure what I'm being asked here...
0
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0answers
22 views

Rotation of axes transformation as definition of vectors

Given a three-axes coordinate system ${1,2,3} $ by the right-hand rule, and a new coordinate system ${1',2',3'}$ , I know that one can define a vector $\vec{x}$ to be something that obeys the ...
0
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3answers
46 views

Calculus and analytic geometry question

Find the tangent of the angle in which the functions $x^3 $, and $x^2 $ intersect $(x≠0)$ . I find this question to be quite funny since the intersection point has two tangents going to it, with ...
0
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2answers
58 views

(I only need some hints)Find the vector equation of a space curve that represents an ellipse with the given center that lies in the given plane

Full disclosure, this is for a Calculus III graded homework set--though we are allowed to use any resources available to complete it. I feel I have a good understanding of space curves, though my ...
1
vote
3answers
25 views

solving for the left side for a double angle formula

$\cos t-\sin 2t=0$ Solve the left hand side so that it equals zero. Do I use $(2\sin t \cos {t})$ for $\sin 2t$?
1
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1answer
149 views

Does a square have an equation? [duplicate]

can you model a square in an equation ? like a circle for example $r^2 = x^2 + y^2$ and lets say we have a square with: centered at $(3,3)$ $2 \leq x \leq 4$ and $2 \leq y\leq 4$ can we somehow ...
1
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2answers
36 views

Finding the point that a normal line goes through

I have been stumped on a homework problem for quite some time and I'm hoping to get some help with it. The line from the origin to the point $(a, f(a))$ on the graph of $f(x) = \frac1{x^2}$ is ...
1
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1answer
28 views

How to Create a Plane Inside A Cube

I have a $e \times e \times e$ cube and I want to create random planes with equation $ax + by + cz + d = 0$ inside this cube. I will put random points on those randomly created planes as well. Here ...
2
votes
1answer
58 views

Proof for $\cos(\alpha)^2 + \cos(\beta)^2 + \cos(\gamma)^2 = 1$ in Euclidean space

What is the proof for this formula: $$ \cos(\alpha)^2 + \cos(\beta)^2 + \cos(\gamma)^2 = 1, $$ where $\alpha$, $\beta$ and $\gamma$ are the angles between a vector and the base of a right-handed ...
-1
votes
1answer
20 views

Graph a line oriented by an specific angle?

I'm writing a software that plots gps data on a map, and so far it has been riddled with complex math problems, many of which I was able to fix by myself but this one I can't figure out. The software ...
2
votes
0answers
40 views

Cauchy-Schwarz on a Euclidian Space

I was thinking about this proof of the cauchy-schwarz inequality, I wanna show that $$|\langle u,v\rangle|\leq|u||v|$$. We know that, $$|\langle u,v\rangle| = ||u||v|\cos{\theta}|$$ where $\theta$ ...
-1
votes
1answer
30 views

locus sections and circles--symmetry

A. Let L = {(x,y,z)|the distance from (x,y,z) to the y-axis is 6}. Describe what shape L is. So far, I have that it's a circle, but I'm not sure how to describe it fully. Would it be a circle that ...
1
vote
1answer
10 views

Finding the degree to cover a flare shape

I have a lamp shade frame that I want to cover with bamboo slats. The top is 12" around and the bottom is 33" around. The distance from the top to the bottom most part is 6". What degree, with minor ...