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

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How to do this problem with vectors?

A plane flies in a direction NW at an airspeed of $141$ km/hr. If the wind at the plane's cruise altitude is blowing with a speed of $100$ km/hr directly from the north, what is the plane's actual ...
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1answer
60 views

Are my answers correct? (finding intercepts, asymptotes, and extrema)

Are my answers correct? a) (0, 4/3) and (2,0) and (-2, 0) b) Horizontal asymptote: $y = 3$, Vertical asymptotes: $x = 3, x = -3$ c) Extremum is at $(0, 4/3)$, maximum.
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46 views

Create a frustum knowing only two edges

You have a 3D frustum. Consider the smallest face the "near" and it's opposite the "far" face. The near and far faces are both perfect rectangles (not squares). No face is axis-aligned. The near ...
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1answer
46 views

boundary of the half-annulus

Question Let C be the boundary of the half-annulus a^2 < (x^2 + y^2) < b^2 and y>=0 in the x-y plane, traversed in the negative direction what is ∫ (5e^(-7x^2) - y^3) dx + x^3 +cosh^2(5y) dy ...
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3answers
52 views

Tangents from a certain point, to a circle?

We have a circle with radius 2, centred on the origin. Find the equation of the lines passing through the point $(0,4)$ which are tangent to the circle. So we have the circle $$x^2 + y^2 = 4$$ ...
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123 views

Finding coordinates of 4th point in a quadilateral

A(1,5), B(4,0), C(-3,-5) are three vertices of a parallelogram ABCD. Find the coordinates of D, the fourth vertex of the parallelogram *What I've Done:*I've created the structure on a number plane. ...
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73 views

Locus Problem .

Prove that the locus of the middle points of all tangents drawn from points on the directrix to the parabola is $y^2(2x+a)=a(3x+a)^2$
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136 views

affine transformation of triangle

$(a,b,c),(a',b',c') \in (\mathbb{R}^2)^3$ triples of pw different points which are not colinear. Show: There's an affine transformation $x \mapsto Ax+s$ with $Aa+s=a',Ab+s=b',Ac+s=c'$ mapping one ...
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1answer
25 views

Altitudes of a hyperbola

I was doing my engineering graphics assignment and I came across this question A cone of base 60mm diameter and slant height 75mm rests with its base on the horizontal plane. It is cut by an ...
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1answer
58 views

hyperbolic orbits, deriving in cartesian coordinates

I was working on this and I wanted to be sure I wasn't too far off. Given: $\frac{\alpha}{r} = 1 + \epsilon \cos \theta$ where $\epsilon$ is eccentricity. Also $\frac{(x + x_0)^2}{A^2} = ...
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2answers
104 views

Using the Invariant Principle to prove a coordinate can't be reached

The problem is as follows: A robot wanders around a 2-dimensional grid. Starting at (0, 0) he is allowed 4 different kinds of step: 1. (+2, -1) 2. (+1, -2) 3. (+1, +1) 4. (-3, 0) He is trying to get ...
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1answer
53 views

Product of the distance from foci to a tangent is a constant

I am supposed to determine what is the result of said product. Given $P(x_0,y_0)$, I need to calculate the distance from the foci to the tangent line that passes through $P$, and then multiply the ...
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1answer
30 views

Prove that the perpendiculars from P,Q,R to BC,CA,AB respectively are also concurrent.

Let $ABC$ and $PQR$ be any two triangles in the same plane,assume that the perpendiculars from the points $A,B,C$ to the sides $QR,RP,PQ$ respectively are concurrent using vector methods or ...
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2answers
478 views

Rotation in 3D (coordinate system transformation)

How do I rotate a point around point [0,0,0] in 3D. In picture I draw specific situation for illustration. At first I know point G[x,y,z] and I will tranfer it on axiz Z, where distance to center is ...
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1answer
343 views

find the equation of the circle passing through the extremities of the diameter of the circle

find the equation of the circle passing through the extremities of the diameter of the circle $x^2 +y^2 +2x-4y-2=0$ $x^2 +y^2 =0$ $x^2 +y^2 -6x-8y-2=0$ I cant understand what the question asks ...
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71 views

find the equation to the circle circumscribing the quadrilateral formed by the straight lines

find the equation to the circle circumscribing the quadrilateral formed by the straight lines $$2x+3y=2$$ $$3x-2y=4$$ $$x+2y=3$$ $$2x-y=3$$ we can see that the first two and the last two are ...
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1answer
2k views

Equation of a circle given two points and tangent line

I am given that $P(-3,-1)$ and $Q(5,3)$ are points of the circle. Also, the line $L:0=x+2y-13$ is tangent to said circle. The objective is to find the equation of the circle. I thought of a way for ...
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2answers
241 views

How to find the orthogonal trajectories of the family of all the circles through the points $(1,1)$ and $(-1,-1)$?

I'm trying to find the orthogonal trajectories of the family of circles through the points $(1,1)$ and $(-1, -1)$. Now such a family can be given by an equation of the form $$ x^2 + y^2 + 2g(x-y) - 2 ...
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1answer
226 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
31 views

When is this line part of this plane..

given is: $ g: \vec x = \frac{1}{15} \begin{pmatrix} 14 \\ 13 \\ 7 \\ \end{pmatrix} + r* \begin{pmatrix}1\\0\\2 \end{pmatrix}$ and $ E: ax_1 * 2x_2 + x_3 = 4 $ What is "a", when $ g \in E $ ? ...
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1answer
82 views

Common Normal Parabola Problem

Prove that two parabolas $y^2=4ax $ and $y^2=4c(x-b)$ cannot have a common normal other than the axis, unless $ b/a-c>2$. I couldn't think of a satisfactory approach. Please Help.
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If a triangle ABC remains always similar to a given triangle,and if the point B always move along a given straight line, find the locus of C.

If a triangle ABC remains always similar to a given triangle,and if the point A is fixed andB always move along a given straight line, find the locus of C. I have been able to solve it for a right ...
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61 views

Proving a parabola property

I need help with this question that i attempted to solve using the equation $y^2=4ax$ : "Prove that on the axis of any parabola there is a certain point which has the property that,if a chord PQ of ...
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1answer
49 views

Vector Projection with respect to another vector

I have learnt about orthogonal projections, but now there is a new problem regarding non orthogonal projections. As seen in the image, given vector d, i would like to project vector v to the line with ...
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3k views

Find the equation of the plane that passes through the line of intersection of the planes…

Find the equation of the plane that passes through the line of intersection of the planes 4x - 2y + z - 3 = 0 and 2x - y + 3z + 1 = 0, and that is perpendicular to the plane 3x + y - z + 7 = 0. This ...
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1answer
81 views

Infinitely many moons, or one ring to bring them all, a limit to bind it?

The Kanagy clan makes its home on a distant planet of mass $M_p$ with $k$ moons. Suppose the moons are identical with mass $m$. Furthermore, these moons share a common circular orbit on an orbital ...
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92 views

Determining vector line equation from two planes.

Determine a vector equation of the line of intersection of the planes p1: 3x - y + 4z - 2 = 0 and p2: x + 6y + 10z + 8 = 0. I know that I can find the cross product of the normals of these vectors ...
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3answers
264 views

How come the cross product of two planes is collinear with the direction vector of the line?

If two planes intersect in a line, explain why the cross product of the normal vectors of the planes is collinear with the direction vector of the line.
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1answer
109 views

Triangle problem (only know 1 side and y-axis coordinates of 2 points)

I have a triangle problem that I am in desperate need of help with with. Here is what I know ... the triangle is mapped on a graph where I only know y-axis coordinates for 2 points, the length of one ...
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2answers
39 views

How to sketch $f(x)= |2x-1|-|1+3x|$

How would you sketch $f(x)= |2x-1|-|1+3x|$? And how would you work out how to sketch it? Please help!
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1answer
26 views

about projection

Let $L$ be the line in $\Bbb R^3$ containing the point $A (0,1,1)$. Specifically, let $L = \{t[0,0,1]+[0,1,1]\mid t\in\Bbb R\}$. Let $U=[1,-1,0]$ The question is to find the point on the line which ...
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1answer
53 views

show that the straight lines $(a^2-3b^2)x^2+8abxy+(b^2-3a^2)y^2=0$ form with the lie $ax+by+c=o$ an equilateral triangle

show that the straight lines $(a^2-3b^2)x^2+8abxy+(b^2-3a^2)y^2=0$ form with the lie ax+by+c=o an equilateral triangle whose area is $\frac{c^2}{\sqrt{3}(a^2+b^2)}$ is there any other way to solve ...
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2answers
114 views

equation representing 2 straight lines

Let us assume this equation is given to us we have to factorize it $$12x^2 +7xy-10y^2+13x+45y-3=0$$ By solving we get that this represents two straight lines. But how to factorize it? Is there a ...
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77 views

Linear separability in a 2×2×2 cube

Essentially, linearly separable points are just those corners that can be cut off with just one slice as marked out by a hyperplane. Can someone please explain how many cases there are for a ...
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3answers
272 views

Why do we believe the equation $ax+by+c=0$ represents a line?

I'm going for quite a weird question here. As we know, the equation in Cartesian coordinates for a line in 2-dimensional Euclidean geometry is of the form $ax+by+c=0$. I'm wondering why do we ...
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619 views

Given middlepoints of a sides of triangle, find vertices

If $(2, 1)$, $(3, 3)$ and $(6, 2)$ are the middlepoints of a sides of a triangle, what are the coordinates of a vertices of a triangle? This part of the book deals with midpoints, with formula: ...
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Two points: Distance, midpoint, equations of line passing through them and perpendicular line [closed]

Consider two points (-2,3) and (-1,6). Calculate the distance between these two points. Find the midpoint. Obtain the equation of the line passing through the given points. Find the equation of the ...
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1answer
168 views

How to show that a line pass through a point?

How to show that a line pass through a point? Two fixed straight line $OX$ and $OY$ are cut by a variable line at the points $A$ and $B$ respectively and $P$ and $Q$ are the feet of the ...
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1answer
49 views

If $l(x,y)=0$ and $l'(x,y)=0$ intersects in $P_0(x_0,y_0)$ then $ \lambda l (x,y) + \lambda ' l'(x,y)=0 , \quad ( \lambda +\lambda ' =1 )$

I'm reading "What is Mathematics?" and found this question. let $l(x,y)=0$ represents the equation $ax+by+c=0$ of a line $l$,and so does $l'(x,y)=0$ . now let $\lambda + \lambda ' =1 $. Show that, if ...
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Define “y” value from the equation of circle

Let's take a circle. It has the following general equation to describe it: $(x-u)^2+(y-v)^2=r^2$ ,where $u,v$ is the coordinates of the center of the circle, and $r$ is the radius of the circle. If ...
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80 views

how do i calculate coordinates of point on rectangle based on angle

If i want to find out what is x and y of point lies of a line drawn from center of the rectangle to outward at certain angle. take a look at picture below.
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$u= \frac {1}{2i} -\frac 3j$ , find a vector perpendicular to u [closed]

Find the interior angle of the triangle ABC give the point $A(3,4), B(-1,-7), C(-8,-2)$. I am a beginner.
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The velocity of a set of axes with respect to themselves

I am reading Whittaker's Analytical Dynamics. This is chapter 8 Non Holonomic Systems. Dissipative systems. Paragraph 88 is Equations of motion relative to axes moving in an arbitrary way. Let $G$ be ...
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1answer
372 views

Ellipse in Cartesian and in Polar Coordinates

So I was studying about ellipses in Polar Coordinates, and the book said Let F be a fixed point, and l be a fixed line in a plane. Let e be a fixed positive number. The set of all points P in ...
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1answer
93 views

Find the locus of the point which divides segment AB internally in the ratio 1:2.

A and B are variable points on X and Y axis respectively,such that l(ab)=4.Find the locus of the point which divides segment AB internally in the ratio 1:2. I think that it must be a circle . or ...
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635 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
73 views

Problem regarding lines (Analytic Geometry)

Here is the problem: The base of a triangle has a fixed position and its length is constant and measures a. The difference of the squares of the other two sides is constant and measures $b^2$. ...
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582 views

How to prove the parallelogram law by using geometric method?

How to prove th parallelogram law by using geometric method? NOT ALGEBRA
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792 views

How to find a perpendicular vector

After thinking longing i can't figure it out no matter what.. So i have 3d line starts (0,0,0) and ends (3.5,3.5,2.5) so therefore has length of about 5. Now how do i find out vector that is ...
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1answer
102 views

Direction cosines

I have two vectors, A and B that meet at point R. The vector RA has a magnitude 2km and direction cosines cos alpha=0.768, cos beta=0.384, cos gamma=0.512. The vector RB has a magnitude 4km and ...