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

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Find the equation of a parabola (in general form)

Find the equation of the parabola with axis parallel to the $y$-axis, passing through $(1/2,-5/2),(3/2,-9/4)$ and $(-7/2,3/2)$.
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Pappus Locus Problem

This is a special case of the Pappus problem or the Pappus Locus Theorem. Let $L_1$, $L_2$, $L_3$, and $L_3$ be $4$ distinct lines in the plane. For a point $p$ in the plane, denote the (orthogonal) ...
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1answer
70 views
+50

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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0answers
25 views

Circle Tangent question

I would like to ask for assiatance on the following: Find the eqation of a circle, with a radius of$\sqrt 2$ , which also has as tangetns the lines: $ y=x+2 $ , $ y=-7x $. It is known that the ...
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1answer
1k views

Finding a vertex of a triangle knowing the other two and its area

I have vertix A, vertix B and the area of a triangle, and I need to find the coordinates of vertex C, knowing that it's on the bisector between the first and the third sector of the Cartesian plane. ...
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1answer
88 views

Tensor notation (practicing)

I'm praticing tensor notation, and I want to prove this way that given vectors $A,B,C,D$ then $(A \times B) \times (C \times D) = \det(A,C,D)B - \det(B,C,D)A$, where $\det$ means the triple product. ...
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1answer
38 views

List of topics for basic calculus (1st,2nd,3rd semester)

I am an computer science student, currently studying in 2nd semester. Therefore my math courses are pretty weak. Although I "aced" them, I still feel I could use some extra basic calculus knowledge in ...
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2answers
192 views

Why is the locus of the centres of the circles passing through two points is the perpendicular bisector of the two points?

Why is the locus of the centres of the circles passing through two points is the perpendicular bisector of the two points?
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3answers
133 views

Analytic Geometry

How does one solve: Find the equation of the circle which has it's center on the line $y= 3-x$ , and which has as tangents the lines $ 2y-x = 22, $ $ 2x+y=11 $ ?
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2answers
33 views

Maximize the distance between a point and a bounding rectangle

There are $n$ random points in the $x-y$ plane, whose coordinates are known beforehand. We can use a minimum bounding rectangle (MBR) to bound these points. In this scenario, the MBR can be rotated, ...
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1answer
45 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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1answer
21 views

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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1answer
23 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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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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0answers
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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4answers
49 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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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
5k views

How do I find the equation of a tangent line to a curve?

I'm given $x^2+2x-4$ at $x=2$ and I have to find the tangent line to this curve at that point...
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2answers
34 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
23 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
250 views

Angle between planes

If the angle between two planes is $\alpha$ , why is the angle between normal of the two planes is $\pi - \alpha$ ? Also Why angle between a line and normal to a plane is $\pi/2 -\alpha$ if angle ...
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3answers
61 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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1answer
206 views

Intersection of two lines

What is the suggested method to find the intersection of two line *segments in 3D space programmatically? I mean there are various methods to solve a set of 2 linear equations, eg. Using ...
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2answers
42 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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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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1answer
39 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
500 views

Find the area of the triangle using analytic geometry

We have a $\triangle ABC$ with: Base $AB$ with length 14 $AC$ with length 15 $BC$ with length 13 Find the area of the triangle using analytic geometry.
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0answers
22 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
59 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
232 views

Find locus of points relating to an ellipse

I would like to find the equation of the following locus. For a big circle C centered at (0,0), the locus of points that the sum of distances to Y-axis and to C is 1, say in the first quadrant, is ...
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1answer
36 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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1answer
175 views

Difficult volume computation inside an ellipsoid and above a plane

I am taking a Calculus course currently and am stuck on the last question of my assignment. Find the volume of the region inside the ellipsoid $\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} ...
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2answers
53 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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3answers
78 views

2 dimensional coordinate geometry

If $L_1$ and $L_2$ are two lines belonging to the family of lines $(3+2s)x+(4+3s)y=7+5s$ such that they are at maximum and minimum distances from the center of the circle $3x^2 +3y^2 -12x-18y-91=0$, ...
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1answer
281 views

Find equations of the ellipses given conditions on the directrices, foci, and vertices

The ellipses have their centers at the origin and their major axes on the $x$-axis. Find the equation: with distance between directrices $27$, and between foci $3$; with a focus at $(-\sqrt{13},0)$ ...
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2answers
29 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
29 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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2answers
39 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
56 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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1answer
31 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
647 views

Find the equation of an ellipse passing thru (-2, 9/4), vertex at (4,0)…

Find the equation of an ellipse a) passing through $(-2, 9/4)$, vertex at $(4,0)$ b) passing through $(0, \sqrt{5})$ eccentricity equal to $\frac{4\sqrt{21}}{21}$ I'm having a hard time to solve ...
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1answer
30 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 ...