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

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3
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0answers
124 views

Calculate the flux of $\underline{v}$ across the boundary of the sector.

For $a\in(0,1)$, calculate without use of the divergence theorem the flux of $\underline{v}(x,y) = g(y/x)(-1/x,1/y)$ across the boundary of the sector $ S_a := \{(x,y)\in \Omega : 1\leqslant x^2+y^2 ...
0
votes
1answer
52 views

Find the equation of the parabola with its vertex on the line $2y-3x=0$?

Its axis of symmetry is parallel to the x-axis, and it passes through the two points $(3,5)$ and $(6,-1)$
0
votes
1answer
30 views

Paramaterizing a Parabola with $3$ points.

Let $A, B, C$ be vectors in $\mathbb R^2$. I want to show that the set $\{A+tB+t^2C\mid t\in\mathbb R\}$ defines a parabola in $\mathbb R^2$, but I'm having a hard time doing so, since I can't solve ...
5
votes
1answer
63 views

What is the equation of the reflections of a fixed point across all the tangents to a fixed circle?

Given a fixed circle "c" and a fixed point "A" (in the plane of the circle), draw the tangent to the circle at a variable point "X" (movable, but constrained to be on the circle), reflect "A" across ...
0
votes
0answers
42 views

What is the equation family of the projectile-motion-with-air-resistance eqn?

The general form of the equations of projectile motion with air resistance are (from here) $s_y(t) = -\frac{mg}{k}t + \frac{m}{k}(v_{yo} + \frac{mg}{k})(1 - e^{-\frac{k}{m}t})$ and $s_x(t) = ...
0
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0answers
25 views

coordinate geometry problems

If $X \cos A +Y \sin A = p \,$ where $p = – \sin^2A/\cos^2A$ be a straight line Prove that perpendiculars on these straight lines from the points $(m^2, 2m)$ , $(Mm, M+m)$ , $(M^2, 2M)$ form a ...
0
votes
0answers
36 views

Unit sphere x axis intersections

This is a problem from a vector calculus textbook, Higher Order Derivatives Consider the unit sphere S given by $x^2+y^2+z^2=1$. S intersects the x-axis at 2 points. Which variables can we solve for ...
0
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0answers
86 views

Asymmetric hyperbola-type curve? (for fitting to data)

I have this question: what would be the name and equation of a curve which resembles a parabola but has not the requirement of symmetry? So the general parabola equation is: $y=ax^2+bx+c$ I must ...
12
votes
3answers
237 views

How to generate points uniformly distributed on the surface of an ellipsoid?

I am trying to find a way to generate random points uniformly distributed on the surface of an ellipsoid. If it was a sphere there is a neat way of doing it: Generate three $N(0,1)$ variables ...
0
votes
0answers
11 views

Homothetic center of two spheres

If the external homothetic center of two circles in the cartesian plane can be found with the following formula: $(x_e, y_e) = \frac{-r_2}{r_1 - r_2}(x_1, y_1) + \frac{r_1}{r_1 - r_2}(x_2, y_2)$ Am ...
0
votes
2answers
69 views

Equation of circle in terms of length of arc above $x$-axis

Say I have a circle centered at $(0,b)$ that passes through $(-5,0)$ and $(5,0)$ and has upper-half length $d.$ Now I've figured out that the equation of the circle is $$x^2 + (y-b)^2 = 5^2 + b^2$$ ...
0
votes
1answer
21 views

Parabola max. number

If the directrix and the tangent at vertex of a parabola are given then what is the maximum number of parabolas that can be drawn? Well according to me the answer should be 1 because the distance ...
4
votes
4answers
101 views

Equation of a line tangent to circumference

Discover the general equation of the tangent line to the circumference $x^2 + y^2 - 2x + 4y + 1 = 0$ by the point $(3,4)$. NO CALCULUS. by the circumference equation i discovered that $C(1, ...
1
vote
2answers
89 views

implicit equation for elliptical torus

I just wondering what the implicit equation would be if an ellipse with major axis a and minor axis b, rotating about the Z axis with a distance of $R_0$. The $R_0$>a and $R_0$>b which means the ...
0
votes
1answer
33 views

How to find the equation of conics given foci and directrices

Find the equations of the following conics, each with its centre at the origin. (a) A hyperbola with foci $(\pm4, 0)$ and directrices $x= \pm2$ (b) An ellipse with foci $(0, \pm4)$ and directrices ...
1
vote
3answers
66 views

Equation of line passing through point.

The straight line $3x + 4y + 5 = 0 $ and $4x - 3y - 10 = 0$ intersect at point $A$. Point $B$ on line $3x + 4y + 5 = 0 $ and point C on line $4x - 3y - 10 = 0$ are such that $d(A,B)=d(A,C)$. Find ...
1
vote
1answer
12 views

geometry of a hyperbola and circle drawn together

how to calculate the radius of a circle which is drawn below(inwards) the hyperbola curve touching it.need a relationship between these hyperbola and circle .If a circular object is place below the ...
0
votes
3answers
55 views

Find a vector minimizing the distance from set

Find a vector $\Pi_Z(x)$ minimizing the distance between $x=(5,10)\in\mathbb{R}^2$ and set $Z=\{(x,y)\in\mathbb{R}^2:x\ge0, y\le\sqrt{x}\}$
0
votes
2answers
37 views

Is there a function whose graph is contained in one quadrant of the coordinate plane?

Is there a function whose graph is contained in one quadrant of the coordinate plane? It should be related to maths and not physics. Please give me the equation and if possible its picture.
2
votes
1answer
41 views

How to find the equation for the line $t$, in the plane $\pi$ and concurrent to other 2 lines

The exercise says that $t$ is in the plane $\pi: x-y+z =0$ and is concurrent to the lines: $$r:\\x+y+2z=2\\x=y$$ and $$s:\\z=x+2\\y=0$$ I've transformed $r$ to the form: $$r:\\x = \lambda\\ y = ...
0
votes
1answer
16 views

Find the line $t$ that is concurrent to $r$ and $s$ and parallel to $MN$

I need to find the vector equation for the line $t$ that is concurrent to both: $$r:X = (1,1,-1)+\lambda(2,1,-1)$$ and $$s:\\x+y-3z = 1\\2x-y-2z=0$$ And also, $t$ is parallel to $MN$ when: $$M = ...
0
votes
1answer
33 views

Find the vector equation of the line parallel to the plane $\pi$, perpendicular to the line $AB$ and that intercepts $s$

I have the plane: $$\pi:2x-y+3z-1 = 0$$ $$A = (1,0,1), B = (0,1,2)$$ And $$s: X = (4,5,0) + \lambda (3,6,1)$$ I need to find a line that is perpendicular to $AB$, parallel to the plane $\pi$ and ...
0
votes
1answer
33 views

Find the equation of the plane that contains the line $r$ and makes an angle with $s$

I have the line: $$r:\\3z-x = 1\\y-1 = 1$$ And the plane makes an angle $\theta = \arccos \frac{2\sqrt{30}}{11}$ with the line: $$s:X = (1,1,0) + \lambda(3,1,1)$$ What I tried: From the equations ...
0
votes
1answer
25 views

Vectors - theory on cross product

If $X$ is a point on a line through $P$ and $Q$, $X=OX, P=OP, Q=OQ$ (all are vectors but $X$) then: $$X \times (Q-P) = P \times Q$$ I subbed in the $OX$, etc and simplified, but did not get each ...
1
vote
1answer
59 views

Why this isn't working? Find the points of the line $r$ that has the distance $\sqrt{\frac{14}{3}}$ from line $s$.

I have the line $$r:\\x+y=2\\x=y+z$$ and $$s:x=y=z+1$$ I need to find the points of $r$ that has distance $\sqrt{\frac{14}{3}}$ from $s$. What I tried: By using the formula for distance of a ...
0
votes
0answers
47 views

Find the symmetric equation of the tangent line to the curve

Find the symmetric equation of the tangent line to the curve defined by the intersection of $3x^2+2y^2+z^2=49$ and $x^2+y^2-z^2=10$ at the point $(3,-3,2)$. I know that when you obtain the partial ...
1
vote
2answers
219 views

How to find the number of squares formed by given lattice points?

Let us say that we are N integer coordinates (x, y) - what would our approach be if we were supposed to find the number of squares we could make from those given n points? Additionally, if we were to ...
0
votes
1answer
25 views

Finding circumcenter

In a triangle A(1,2) B(2,3) C(3,1) and $\angle A = cos^{-1}(4/5)$, $\angle B = \angle C = cos^{-1}(\frac{1}{\sqrt{10} }) $ Ordinate of circumcentre of the $\triangle$ is ? I have tried solving by ...
2
votes
1answer
60 views

Find the maximum possible area for the triangle

Two vertices of an isosceles triangle are (1,2) and (4,6). The inradius of the triangle is $\frac{3}{2}$. Find the maximum possible area for the triangle. My work, for the two possible structures of ...
1
vote
1answer
26 views

equation of major axis of an ellipsoid

What is the equation of 3 major axes of the following ellipsoid? $$ \begin{pmatrix}x & y & z\end{pmatrix} \begin{pmatrix} \alpha_1 & \beta_3 & \beta_2\\ \beta_3 & \alpha_2 & ...
-1
votes
4answers
268 views

Find equation of a line perpendicular to the tangent of curve at a given point.

I need to find the equation to the line perpendicular to the tangent to the curve $y = x^3 -3x +1$, at the point $(2,3)$. Our teacher assigned us homework on stuff we haven't learned, so please if ...
1
vote
1answer
29 views

How does one obtain Hesse normal form of plane equation?

We have been studying the Hesse normal form of the plane equation, but the sketch of the plane in space given by the lecturer was horrible. Basically I ask you to explain me how does one obtain the ...
0
votes
0answers
31 views

What is the graph of the identity function in cartesian coordinates?

Why is that the graph of f(x) = x is the straight line that is the bisector of the first quadrant? (or, amounting to the same thing, the bisector of the third quadrant) By calculating the outputs for ...
1
vote
1answer
45 views

What's the parametric equation for the general form of an ellipse rotated by any amount? Thanks.

What's the parametric equation for the general form of an ellipse rotated by any amount? Preferably, as a computer scientist, how can this equation be derived from the three variables: coordinate of ...
0
votes
0answers
32 views

Finding an ellipsoid equation

I want to find 3D equation of a falling droplet that I have considered it as an ellipsoid. I put two cameras, one in xy plane and another in zy plane to capture two projected views of the droplet and ...
1
vote
2answers
64 views

Distance of a Point from Hyperbola

Consider the part of hyperbola $H_{+}=\{(x,1/x)\colon x>0\}$ in the first quadrant, and $(a,b)$ any point in the plane (for sake of convenience, say $a,b>0$). If $(a,b)$ does not lie on the ...
1
vote
1answer
41 views

Locus of a point that satisfy a condition on the square of distances to two lines and their intersection

Find the locus of a point such that the square of its distance to the point of intersection of two perpendicular lines is equal to the sum of its distances to those lines. Assume $P(x,y)$ is any ...
0
votes
1answer
19 views

I want to know where I did wrong in finding the plane equation

I am asked to give 3 plane equation where the third plane will passes through the intersection of the first 2 planes and parallel to y axis. I came up with 2 plane equation which is also parallel to ...
2
votes
1answer
43 views

Geometric locus

The problem is: Let $A$, $B$ and $C$ be fixed points, and $α,β,γ$ and $κ$ are given constants, then the locus of a point $P$ that satisfies the equation $$α(AP)^2+β(BP)^2+γ(CP)^2=\kappa,$$ is a ...
0
votes
2answers
20 views

Describing vector equation geometrically

How would I describe geometrically the vector equation: $$\vec{x} = s(0,2,1) + t(1,1,-1) ,\qquad s,t \in \Bbb R$$
2
votes
2answers
42 views

find equations of an ellipsoid axes

I have an ellipsoid with the center point at the Origin and the following equation: $$\alpha_1 x^2+\alpha_2 y^2+\alpha_3 z^2+2\beta_1 zy+2\beta_2 xz+2\beta_3 xy=1$$ How can I find the equations of ...
0
votes
0answers
25 views

when you draw 1 altitude/ perpendicular bisector of an equilateral triangle, what can you form?

when you draw 1 altitude/ perpendicular bisector of an equilateral triangle, what can you form such that when you draw 4 equilateral triangles the foot of the perpendicular of the equilateral ...
0
votes
1answer
18 views

Plotting Particular Conic Section

How would I plot $-2x^2 -2y^2 = 1$ on the x-y plane ? I believe it is an ellipse, since the coefficients have the same sign, I just don't know what the major and minor axes would be nor how to plot.
1
vote
1answer
81 views

General solution for intersection of line and circle

If the equation for a circle is $|c-x|^2 = r^2$ and the equation for the line is $n \cdot x=d $, and assuming that the circle and line intersect in two points, how can I find these points? Also as ...
0
votes
0answers
16 views

How to insert a simplifier assumption in our equations set to find an ellipsoid equation

Regarding the below question: Finding equation of an ellipsoid two projected views (two ellipses) is not enough to solve the equation set and find a unique ellipsoid. For example, I chose a ...
0
votes
2answers
34 views

What is the d in the formula of a plane in $ R^3$

In algebra the formula for a line is $y=ax+b$ the $b$ moves the position of the line up and down the y axis. The formula for a plane is given to me as $ax+by+cz+d=0$ the $d$ must move the position of ...
1
vote
1answer
41 views

Using substitution to determine if a given point is on the line

Is it necessary to rearrange the equation of a line so that it is in the $y=mx+b$ form before using substitution to check whether a point is on the line? If yes, why? If no, why?
1
vote
1answer
91 views

Finding equation of tangent of a circle that intersects the origin?

Given: circle with equation $(x-2)^2+(y-1)^2=4$. How to find equation of tangent line to the circle that intersects the origin? I easily found out that one of the tangents is $x=0$.
1
vote
1answer
349 views

Determine y-coordinate of a 3rd point from 2 given points and an x-coordinate.

I'm working through the "Calculus 1" Coursera course (offline version, so no forums) and am struggling with the following question in the topic on Limits: Consider points A=(-10,-4) and C=(8,5). ...
-1
votes
2answers
70 views

Find the other 2 points of a rectangle? [closed]

$PQRS$ is a rectangle with vertices $P(-4,-1)$ and $Q(-6,5),$ and $PQ=2(QR).$ Find the coordinates of $R$ and $S$? I'm so stuck please help! There are 2 answers for each point. almagest has the right ...