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

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Check my solution to this trig inequality

Problem $1.88$ : Solve $$\cos x\lt \frac{\sqrt{3}}{2},\qquad x \in [0,2\pi]$$ I found the set of solutions to be $S=[0,2\pi]-\left[\dfrac{\pi}{6},\dfrac{11\pi}{6}\right]$ Is this correct? Thank you.
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0answers
29 views

Book suggestions on projective geometry

I want to be acquainted with projective geometry, so I'm asking for a reference. I need some words to explain my specific background and motivation. There are many things I learnt related to ...
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1answer
21 views

PROOFING: Show that the area of the shape can be written as $A=200r-r^2 (2+ \pi/2)$

A $\rm200\,m\,$ fence is to placed around a lawn of this shape. We know that $x$ in terms of $r$ : $$x=100-\dfrac{(2+\pi)r}2$$ How do I show that the area of the lawn, $A$, can be written as: ...
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1answer
31 views

Tangent at a singular point

I'm looking at this question If the tangent at the point $P$ with coordinates $(h, k)$ on the curve $y^2 = 2x^3$ is perpendicular to the line $4x = 3y$, find $(h, k).$ This is how I attempted it ...
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0answers
18 views

Find coordinates for point on circle

We got two points on circle and we know about center angle between two points.If we know coordinates of center $(x_1,y_1)$ and point $A=(x_2, y_2)$ and angle $\alpha$, how can we find the coordinates ...
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4 views

Mirror a line over a plane

I am trying to mirror a line over a plane, but I am not sure if I am doing it right, so please tell me if something that I do is wrong. I have 2 points $A(1, 2, 1);B(-1,0,2)$ and I have to mirror the ...
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2answers
34 views

Shortest distance form a point to a plane [closed]

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 $P_0$.
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1answer
24 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 ...
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2answers
31 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?
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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?
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2answers
38 views

Orthogonal tangents to an ellipse [duplicate]

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 ...
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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
47 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 ...
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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 ...
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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}?
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1answer
38 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!
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1answer
55 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. ...
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1answer
35 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 ...
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1answer
21 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 ...
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2answers
27 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 ...
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2answers
52 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 ...
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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 ...
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2answers
37 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} = ...
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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 ...
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2answers
28 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 ...
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1answer
30 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).
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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 ...
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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 ...
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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 ...
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63 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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30 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 + ...
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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: $$ ...
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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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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 ...
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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 ...
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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 ...
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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| = ...
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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: ...
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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 ...
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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...
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23 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 ...
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3answers
47 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 ...
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60 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 ...
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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 $\sin 2t$?
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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 ...
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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 ...