Questions tagged [locus]

For problems that involve a specific set of locations of points. Locus is an important part of the coordinate geometry. In geometry, a locus (plural: loci) is a set of all points (commonly, a line, a line segment, a curve or a surface), whose location satisfies or is determined by one or more specified conditions.

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7
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1answer
198 views

Show that the locus of a point in this geometric construction is a conic

Start with a circle $c$ (black), conic $d$ (green) and a point $A$. $K$ is a point on the conic, and the tangent at $K$ intersects $c$ at $F$ and $G$. Line $GH$ is perpendicular to $AG$ and line $FH$...
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19 views

An interesting Cardioid-looking locus with hyperbola-looking equation

As we clearly know, the locus of all complex numbers $z$ such that $(|z-z_1|-|z-z_2|)^2=c^2$ is a hyperbola with the foci set on $z_1,z_2$ and with constant value $c$. This can also be written is ...
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1answer
34 views

Given a directrix and a focus, can we use an arbitrary curve as the directrix?

A parabola is defined as the set of points equidistant from a directrix line and a focus point. But what if we allow the directrix to be an arbitrary curve, including possibly another parabola? What ...
5
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4answers
194 views

What is the locus of $Z = \frac{4-z}{4+z}$ if $|z|=4$?

The question is: Given $$Z=\dfrac{4-z}{4+z},$$ if $|z|=4$, then find the locus of $Z$. The solution in the textbook is: $$Z=\dfrac{4-z}{4+z}\iff z=\dfrac{4(1-Z)}{1+Z},$$ so, $|z|=\left|\frac{4(1-Z)}{1+...
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47 views

Proving equivalency of locus and described curve

The locus of a geometric figure can be obtained by encapsulating the known properties of the figure in some parametric equations, and then combining them into a single equation. As I understand it, ...
2
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1answer
137 views

How do I know when squaring is valid or not?

Question: The straight-line $x\cos\alpha+y\sin\alpha=p$ intersects the x & y axes at A & B respectively. Considering $\alpha$ as a variable, show that the equation of the locus that the middle ...
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2answers
198 views

A cube ABCDEFGH is given. Determine the locus of all midpoints of segments MN, where M is any point on segment AC and N any point on segment FH.

Question is:A cube $ABCDEFGH$ is given. Determine the locus of all midpoints of segments $MN$, where $M$ is any point on segment $AC$ and $N$ any point on segment $FH$. My understanding : Here $M$ and ...
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1answer
92 views

Are there circular limaçons?

Trying to get a general definition for auxiliary circles in the context of conic sections (Wikipedia redirects me to just 'Ellipse'), I came to this table at the wiki for pedal curves. It states that ...
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2answers
52 views

Locus of a moving point when tan of half look angle is inversely proportional to distance.

A point $ P$ moves such that the tan of bisected angle $\alpha $ subtended between two fixed points $F_1$ and $F_2$ at $P$ is inversely proportional to distance $d$ of $P$ to line $F_1-F_2$. Show ...
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0answers
18 views

Understanding about locus

As I have been studying about locus,then I came across its definition as Locus is a set of points which satisfies a given condition or property (law/equation/rule) We have been taught about some locus ...
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23 views

How is the motion of the point whose position vector satisfies this differential equation confined to one plane?

Let $P$ be a point moving in three dimentional space in such a way that its position vector $\mathbf{r} = \vec{OP}$ relative to the origin $O$ of a cartesian rectangular coordinate system $Oxyz$ ...
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2answers
66 views

A faster way to find the complex locus of $ | z^* + 2i | + | z | = a $

I need to find the equation of the locus of $ z = x + iy $ which satisfies $$ | z^* + 2i | + | z | = a $$ for some real constant $ a $, where $ z^* $ is the conjugate, $ x - iy $. I already solved ...
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1answer
34 views

Find the area of the locus of center of circle formed given the following conditions

$A$ is a fixed point on a circle of radius $1$ unit, $P$ is a variable point on this circle. A circle is formed touching the $CP,CA$ and the $Arc(AP)$ where $C$ is the center of the original circle. ...
0
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1answer
29 views

How do I find the half-line equation for the locus $\operatorname{arg}(z - a) = \frac{-3\pi}{4}$?

So there are a locus composed of points $m$ that is $z$ and it's a half-line: $\operatorname{arg}(z - a) = \frac{-3\pi}{4}$ $a = - 1 + i$ The problem is that I have no idea how to find out the ...
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3answers
78 views

Understanding how the equation $(x-6)^2+(y-9)^2+ \lambda (3x-y-9)=0$ allows us to find the circle through $(0,1)$ tangent to $3x-y-9=0$ at $(6,9)$

I'm required to find the equation of the circle passing through (0, 1) and touching the line $3x-y-9=0$ at $(6,9)$. Apparently, this is easily done by writing it out as a second degree equation; $$(x-...
7
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3answers
137 views

Step question, locus of points where angle of elevation to tops of flagpoles is always the same

The smooth and level parade ground of the First Ruritanian Infantry Divison is ornamented by two tall vertical flagpole of heights $h_1$ and $h_2$ a distance d apart. As part of an initiative test a ...
2
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1answer
51 views

What is the locus of midpoints of the chords of contact of$ x^2+y^2=a^2$ from the points on the $\ell x + my + n = 0$

What is the locus of midpoints of the chords of contact of $ x^2+y^2=a^2$ from the points on the $\ell x + my + n = 0$ My approach is as follow $\ell x + my + n = 0 \Rightarrow my = - \ell x - n \...
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0answers
57 views

$ABC$ is a triangle. Find the locus of $P$ if $PA^2 + PB^2 = PC^2$ [duplicate]

The exact question is $ABC$ is a triangle. Find the locus of $P$ if $PA^2 + PB^2 = PC^2$. I have solved similar questions in Coordinate geometry where coordinates of the sides were given and I know ...
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1answer
31 views

How can you find the locus of a point that is the same distance from a exponential function?

Okay so this is a question that I came up with few weeks ago, and I still cannot find a answer to this problem. Abstract idea: If you draw a line that is perpendicular to the tangent line of a point $(...
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3answers
83 views

Find the equation of the locus of the mid point of AB as m varies

I am working through a pure maths book as a hobby. This question puzzles me. The line y=mx intersects the curve $y=x^2-1$ at the points A and B. Find the equation of the locus of the mid point of AB ...
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1answer
25 views

Find the equation of the locus of the point P as it moves equidistant from the lines $x = 1$ and $y = 1$

Find the equation of the locus of the point P as it moves equidistant from the lines $x = 1$ and $y = 1$ I cannot see where I am going wrong here. I say, $|x-1| = |y-1|$ $(x-1)^2 = (y-1)^2\implies x^2-...
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3answers
220 views

Geometry problem proving that all lines $DE$ passes through one point

Let $I$ is the incenter of $\triangle ABC$. Let $K$ be the circumcircle of $ABC$. Let $D$ be a variable point on arc $AB$ on $K$ not containing $C$. Let $E$ be a point on line segment $BC$ such that $\...
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1answer
57 views

Locus of complex number z

Let $λ,z_1,z_2,z_3 ∈ \mathbb{C}$ (Complex Numbers) are such that $$\frac{z_3-z_1}{z_2-z_1} = \lambda.$$ Now if $$λ=e^{it}$$ where $t ∈ \mathbb{R}$ and $z_2, z_3$ are fixed, then what will be the ...
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2answers
44 views

Motion of particle with acceleration $\hat\theta$

Suppose a particle has acceleration $a=-\hat\theta$ , how can I find its trajectory?( for simplicity lets take $v=0$) I tried to convert $\hat\theta$ into cartesian ,but the expression turned to be ...
0
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1answer
61 views

What would the locus of a point be whose difference of distances from any two points is always equal to the distance between the points?

Let me give an example to explain my question more, what would the collection of possible points which satisfy the following condition look like? : the point's distance from $(-2,0)$ and $(2,0)$ is ...
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1answer
102 views

what is the complex number that satisfy both loci $\lvert z\rvert=4$ and $\lvert z+2\rvert=\lvert z-4\rvert$?

I have solved most of the question. This is my work but I can't seem to find what this part is. It's supposed to be $1\pm i\sqrt{15}$ but I don't know why. Any help is much appreciated. Thank you ...
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0answers
29 views

Egguation of this locus

Recently I saw this YouTube video where an egg shape is drawn using a construction similar to that of an ellipse. So I was wondering about the equation of this shape. Can we describe it by just using ...
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2answers
56 views

Locus of a point of intersecting lines

The perpendicular from the centre of the ellipse $$b^2x^2+a^2y^2=a^2b^2$$ to the tangent at any point $P$ meets the line joining  point $P$ to the focus $(ae,0)$ at $Q$. Prove that the locus of $Q$ is ...
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0answers
53 views

The locus of points with respect to the ellipse.

The chord of contact of the point $$(x_1,y_1)$$ with respect to the ellipse $$b^2x^2+a^2y^2=a^2b^2$$ cuts the axes at $L$ and $M$. If the locus of the mid-point of $LM$ is the circle $$x^2+y^2=1$$, ...
2
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1answer
40 views

What curves subtend a constant fraction of angle at center.

It is known that if the angle subtended from two foci $ \text{ in the sketch} (\pm 3, -4) $ at origin and at the arc are $ (\gamma, \gamma/2 ) $ respectively then the curve is a circular arc. What ...
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0answers
25 views

locus of the points between two spirals

Suppose you have two spirals, whose equations in parametric form differ only by the radius, which in the second one is shifted by +1: $t \in \mathbb R^+$ spiral #1 : $x = t \cdot cos(t); y = t \cdot ...
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1answer
61 views

Find a locus of a centroid point of a triangle.

When doing another geometry proof, I have this problem: Let $A$ is the point outside the circle $(O;R)$ such that $OA=2R$. The line $d$ goes through $A$ and cut the circle at $2$ distinct points $E$ ...
1
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1answer
64 views

Construct the vertex of an angle given three points on the plane.

Suppose you are given three points A,B,C on the plane. A is on the bisector of angle XOY, B is on the side OX and C is on the side OY. How can you find the locus of the vertices(find all possile ...
4
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3answers
153 views

Find the locus of this point P

Find the locus of all points inside $\triangle ABC$ such that $PA^{2}+PB^{2}=PC^{2}$. At first, i tried finding a right angled triangle and then tried to go on applying Pythagorean Theorem and finding ...
0
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1answer
56 views

Locus of the intersection point of angle bisectors of angles made by 2 fixed lines

"$AB$ and $CD$ are two fixed straight lines and a variable straight line cuts them at $X$ and $Y$ respectively. The angular bisectors of $\angle AXY$ and $\angle CXY$ meet at $P$. Find the locus ...
0
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1answer
26 views

find the locus of points M

find the locus of points M, the difference between the squares of the distances from which to two given points A and B is equal to a given value C, At which C does the problem have a solution? Anyone ...
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5answers
88 views

Finding the locus of $b^2-2x^2=2xy+y^2$

I was asked to determine the locus of the equation $$b^2-2x^2=2xy+y^2$$ This is my work: Add $x^2$ to both sides: $$\begin{align} b^2-x^2 &=2xy+y^2+x^2\\ b^2-x^2 &=\left(x+y\right)^2 \end{...
2
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2answers
105 views

Contruction of a triangle, knowing one angle and it perimeter. What is the locus of a point which splits the perimeter in two at a specific length?

Building a triangle (ABC) with a fixed perimeter (P) knowing just one angle (Summit A) is not that easy. Having fixed one adjacent side (segment AB), we know that the sum of both the others sides is a ...
2
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2answers
112 views

Prove the locus of $M$ is a conic passing though $I$ and $J$.

Problem: In the perspective plane $\mathbb{P}^2$, given a conic $(S)$ and a line $d$ having no intersection with $(S)$. Fixed two points $I$ and $J$ on $d$. Take a variety point $M$ such that the ...
0
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2answers
56 views

PQ subtends an angle of $45^{\circ}$ to the origin. Find the locus of the mid point of a chord PQ to circle $x^2+y^2-2ax-2by=0$

A chord PQ of the circle $x^2+y^2-2ax-2by=0$ subtends an angle of $45^{\circ}$ to the origin. Given that M is the mid point of PQ. Find the equation of the locus of M. MY ATTEMPT Centre of the given ...
0
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0answers
49 views

What is the locus of points $(s,t)$ equidistant from the convex side of parabola $y = x^2$? [duplicate]

What is the locus of points $(s,t)$ equidistant from the convex side of parabola $y = x^2$? Now, it is quite easy to determine s and t separately as functions of x, but can you define t as a function ...
0
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1answer
24 views

Ambiguity in the center of the locus of $|z-z_o|=r$

Let $z_o$ be a fixed complex number and $z$ be a variable complex number I've been told that the locus of $z$ in $|z-z_o|=r$ where $r\gt0$ is a circle centred at $z_o$ I have not been able to ...
0
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2answers
228 views

Midpoint of a line segment passing through two points.

Let $C_1$ be a circle defined with $X(-2,7)$ and $Y(2,-5)$ as the endpoints of the diameter of the circle. Let $C_2$ be a circle defined with $Y(2,-5)$ and $Z(4,-11)$ as the endpoints of the diameter ...
5
votes
4answers
323 views

Let $A=(-1,0),B=(1,0), C$ be points in $\mathbb{R}^2.$ What is the locus of points $\{s\in\mathbb{R}^2:C$ lies on the angle bisector of $AsB$}?

Let $A=(-1,0),\ B=(1,0),\ C$ be points in $\mathbb{R}^2.$ What (shape(s)) is the locus of points $L(C)=\{s\in\mathbb{R}^2:C$ lies on the angle bisector of $AsB$ } ? Obviously the set $L$ depends on ...
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1answer
52 views

Find the locus of points on a complex plane that satisfy |z-3| - |z+3| = 4 [closed]

Can you find the locus of points on the complex plane represented by the equation below: |z-3| - |z+3| = 4
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1answer
49 views

Geometric proof on three colinear points

The following question consists of a final exam, which is originally written in Spanish, so please excuse me if something is lost in traslation. The problem goes as follows: We start of with a rhombus ...
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0answers
31 views

Finding locus of point satisfying given algebraic conditions

Consider $O$ as the origin and a variable straight line drawn through it to cut $a_{1}x+b_{1}y+1=0$ and $a_{2}+b_{2}y+1=0$ in $L$ and $M$ respectively. Let $N$ be a point on the variable line. Find ...
1
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3answers
101 views

Locus of $z$ satisfying $\arg \left(\frac{z-8}{z-2}\right)=\frac{\pi}{2}$

For a given complex number, $z$, find the locus of points on the Argand diagram such that $$\arg \left(\frac{z-8}{z-2}\right)=\frac{\pi}{2}$$ This is my approach: $$\arg(z-8)-\arg(z-2)=\frac{\pi}{2}$$...
3
votes
2answers
101 views

Locus of point at a fixed distance from midpoint of intercepts of a variable line segment with fixed distance

Let A and B be variable points on the x-axis and y-axis respectively such that the line segment AB is in the first quadrant and of a fixed length 2d. Let C be the mid-point of AB and P be a point such ...
1
vote
1answer
327 views

Show that the tangents at P & Q meet on the curve...

A straight line is drawn parallel to the conjugate axis of a hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ to meet it and the conjugate hyperbola in the points P & Q. Show that the tangents at P &...

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