Skip to main content

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.

Filter by
Sorted by
Tagged with
0 votes
1 answer
53 views

Locus of a point whose distance from two points is fixed (but not necessarily equal) in 3D geometry

Suppose there are two fixed points $S_1$ and $S_2$ Let the moving point be $P$ $PS_1$ and $PS_2$ are fixed but not necessarily equal. Now I think it is a circle. Obtained by rotation of vertex of ...
Aurelius's user avatar
  • 471
0 votes
1 answer
38 views

Loci of $C$ such that $AC=2BC$, given that $A$ and $B$ are fixed points

I was solving this complex numbers problem: Find the loci of $|z+1|=2|z-1|$. I solved this problem by substituting $z=x+iy$ and after some manipulations the answer that I got is the circle with ...
Etemon's user avatar
  • 6,782
0 votes
2 answers
24 views

Converting a set of parametric coordinates to Cartesian coordinates

I have the parametric set of coodinates P = $(\dfrac{1+at^{2}+6at}{4},\dfrac{3at^{2}+2at+3}{4})$ , where $a = \dfrac{1}{16}$, after solving a problem involving the locus of a point. I wish to convert ...
Bongo Man's user avatar
  • 331
0 votes
0 answers
18 views

Locus of center of hyperbola that is tangent to coordinate axes

Given the generic hyperbola $ \dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1 $ which is centered at the origin, suppose we shift it and rotate it, such that one of its branches becomes tangent to the ...
Quadrics's user avatar
  • 24.3k
2 votes
2 answers
95 views

How to show that $\left\{z \in \mathbb{C} : \left|\frac{z - 1}{z + 1}\right| < \frac{1}{2}\right\}$ is a disk [duplicate]

I have to prove that the equation $$(z + 1)e^{-z} = 2(z - 1)$$ has only one solution in the open right half-plane. I know that if $z$ is a solution such that $\Re(z) > 0$ then $\left|\frac{z - 1}{z ...
Cyclotomic Manolo's user avatar
0 votes
2 answers
55 views

A curve intersected by a straight line having constant harmonic mean segments

It is known that the if the product of two line segments OA,OB drawn from any point O to a curve is a constant, then the curve is a circle (black). The product is the square of the geometric mean. ...
Narasimham's user avatar
2 votes
3 answers
75 views

Determining the curve to which variable line segments are tangent

A variable line passing through a fixed point $P(x_0,y_0)$ intersects the circle $x^2+y^2=a^2$ at some point $Q$. If all the perpendicular bisectors of the variable line segment $PQ$ are tangent to a ...
Cognoscenti's user avatar
7 votes
4 answers
748 views

Find the center of all circles that touch the $x$-axis and a circle centered at the origin

Given a circle $C$ of radius $1$ centered at the origin, I want to determine the locus of the centers of all circles that touch $C$ and the $x$-axis. This is the red curve in the following Desmos ...
emacs drives me nuts's user avatar
0 votes
1 answer
51 views

Locus of point with constant cosine of angles ratio

How do we find locus of P looking at two foci distance $a$ apart such that $$ \cos \phi/\cos \theta =- k~? $$ (k positive real). Vaguely remember it comes from Electromagnetic theory. Thanks in ...
Narasimham's user avatar
-1 votes
1 answer
39 views

circles with fixed radical axis [closed]

Assume that $l$ is a line and $A,B,C,D$ are points in general position. How many pairs of circles $w_1,w_2$ are there such that $w_1$ goes through $A,B$, $w_2$ goes through $C,D$ and the radical axis ...
ALi1373's user avatar
  • 31
1 vote
1 answer
150 views

Locus of centroid of equilateral triangle inscribed in ellipse.

Problem Find the locus of the centroid of an equilateral triangle inscribed in the ellipse $x^2 / a^2 + y^2 / b^2 = 1$ My attempt I assumed 3 parametric points on ellipse P, Q and R. And assumed the ...
Mark's user avatar
  • 13
1 vote
1 answer
51 views

Concept issue regarding the locus defined by $\arg\left(\frac{z-1}{z}\right) = \frac{\pi}{2}$

I had this simple statement: $$\arg\left(\frac{z-1}{z}\right) = \frac{\pi}{2}$$ I was interested in the locus of $z$. What my understanding is angle between the line segment joining $\overline{OA}$ ...
A shubh's user avatar
  • 183
1 vote
1 answer
57 views

Reflection of a continuous differentiable curve about a line

Let $g(x)$ be the reflection of the continuous and differentiable curve $f(x)$ about the line $x\cos(\theta)+y\sin(\theta)=r$. Find $g(x)$. Also, examine the case in which a curve $f(x)$ is reflected ...
Cognoscenti's user avatar
3 votes
1 answer
333 views

Locus of the free end of a string whirled in a manner as to keep speed constant.

A bob of mass $m$ and negligible radius is tied to a massless string of length $L$ and rotated uniformly along a vertical circle of radius $L$ under the influence of gravity. The free end of the taut ...
Doge with shades's user avatar
6 votes
3 answers
199 views

Generation of a cardioid as the locus of intersection point of two tangents to two intersecting circles

About 8 months ago I came up with this nice feature using GeoGebra but I couldn't prove it, any help would be appreciated We have two intersecting circles and we drew the moving line passing through ...
زكريا حسناوي's user avatar
1 vote
0 answers
31 views

Prove that the locus under given conditions is a helix(Why is path helical when acceleration is cross product of velocity with a given fixed vector)?

So I was solving this larger question and after solving everything, I obtained these data. All derivatives are wrt time unless specified. $f(P)$ represents the position vector of the point $$f(P)=x\...
Aurelius's user avatar
  • 471
2 votes
0 answers
19 views

Proving an identity of distances about tangent of a locus similar to conchoid

Let $l$ be a line and $A$ be a fixed point. Draw a line through $A$ meeting $l$ at $B$. Take the point $C$ on the half-line $BA$ such that $BC$ equals a given constant. Draw the locus of $C$ (called ...
hbghlyj's user avatar
  • 3,045
0 votes
0 answers
21 views

What constraints produce a locus describing a circle where $AB$ is a chord of measure $\gamma$?

Let points $A,B$ be fixed, and let $C$ vary, with the constraint that $\gamma := \angle ACB$ is fixed. It is well known that the locus of $C$ is a double arc, where $AB$ is a chord of measure $2 * \...
SRobertJames's user avatar
  • 4,450
2 votes
1 answer
140 views

Locus of trirectangular tetrahedron [closed]

Let $A,B,C$ be points on the unit circle in the $xy$ plane, and $P$ be a point in space such that $PABC$ is a trirectangular tetrahedron with $P$ at the vertex. Find the locus of $P$ as $A,B,C$ vary ...
godlification's user avatar
1 vote
0 answers
37 views

What is the locus when the radius is cut to its projection?

Fix a circle $\bigcirc O$ and a diameter $d$. For each point $A$ on $\bigcirc O$, let $A'$ be its projection onto $d$, and mark $\Lambda$ on ray $OA$ such that $O\Lambda \cong AA'$. What is the ...
SRobertJames's user avatar
  • 4,450
1 vote
2 answers
146 views

What is the locus of midpoints of chords which pass through point $P$?

Given a fixed circle, find the locus of midpoints of its chords which include a fixed point $P$ Source: Hadamard's Geometry My proof is below, to which I request verification and critique; in ...
SRobertJames's user avatar
  • 4,450
0 votes
1 answer
61 views

With which sign should the modulus function be opened when a point lies below the line? - HW

Question: ABC is an equilateral triangle with A(0, 0) and B(a, 0) (a > 0). L, M, and N are the foot of the perpendiculars drawn from a point P to the sides AB, BC, and CA, respectively. If P lies ...
sanaya's user avatar
  • 11
1 vote
2 answers
122 views

Locus of vector endpoint

Let u and v be two vectors, with their starting point at the center O of the square ABCD, and their endpoints moving along the sides of the square. Determine the locus of the endpoint of u + v! This ...
Birgitt's user avatar
  • 325
1 vote
0 answers
71 views

Find the equation of the locus of a point the difference of whose distances from two fixed points is constant given their coordinates.

So the fixed points are $$F_1=(p_1,q_1)$$$$F_2=(p_2, q_2)$$ Mid-point of foci(centre) is $$\left(\cfrac{p_1+p_2}{2},\cfrac{q_1+q_2}{2}\right)=(c_x,c_y)$$ and the the point $P=(h,k)$ The equation is ...
Aurelius's user avatar
  • 471
2 votes
2 answers
88 views

The midline of a triangle

Triangle $ABC$ is isosceles with $AB = AC$. $P$ is a variable point on $AB$, and $Q$ is a variable point on $AC$, so that $BP = AQ$. Let $O$ be the midpoint of $PQ$. Prove that $d(O, BC)$ is constant, ...
smthBag's user avatar
  • 31
0 votes
1 answer
84 views

A is a given point and P is any point on a given straight line. If AQ = AP and AQ makes a constant angle with AP, find the locus of Q.

I've been thinking a lot about this question for a while now, I checked various books on how one can find the locus of something, but I just can't understand. This is not a "homework question&...
Godot Roy's user avatar
2 votes
2 answers
348 views

Locus of midpoint of chord of an ellipse whose length is constant

Consider an ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2}=1$ We want to find the locus of the midpoints of all those chords whose length is constant $(=2c)$ Here's my approach: Let the midpoint of the ...
Ayush Kumar's user avatar
1 vote
3 answers
203 views

Finding $z \in \mathbb{C}$, with $z \neq -1, i$, such that $\operatorname{arg}(z+1)=\operatorname{arg}(z-i)$.

So for most locus questions, the algebra is fairly straightforward, I usually just substitute $z = x + yi$ into $z$ and see where that takes me. But for this question, I can't seem to get anywhere ...
Crocophobia's user avatar
1 vote
1 answer
75 views

Points $A,B,C$ are given, find a line and a point such that distance of $A,B,C$ from that line is equal to distance of $A,B,C$ from that point.

In the name of God Suppose points $A(1,2)$ and $B(2,5) , C(3,10)$ are given. Find equation of line $l$ and point $F$ such that $AH = AF , BH' = BF , CH'' = CF$, where $AH, BH' $and$ CH''$ are ...
madfd adfd's user avatar
1 vote
2 answers
105 views

$z_1, z_2, z_3\in\mathbb C$ satisfy $|z_1-3|=|z_2-3|=|z_3-3|$ and $\arg(\frac{z_3-z_1}{z_2-z_1})=\pi/6$, compute $z_2^2+z_3^2-3z_2-3z_3-z_2z_3+10$

let $z_{1}, z_{2}, z_{3}$ be three complex number such that $|z_{1}-3|=|z_{2}-3|=|z_{3}-3|$ and $\arg(\frac{z_{3}-z_{1}}{z_{2}-z_{1}})=\pi/6$ then what is the value of $z_{2}^2+z_{3}^2-3z_{2}-3z_{3}-...
Patrick Schick's user avatar
2 votes
2 answers
70 views

Parabolas and Loci

There's a parabola $y = 4x^2$. The point P moves along the curve. A line passing through P and the point $(0, 1)$ intersects the curve again at Q. The tangents to the curve at P and Q intersect at X. ...
Laksh Sharma's user avatar
2 votes
3 answers
118 views

Find locus of all points that its sum of squares of distance between them and point $A(x_A,y_A)$ and $B(x_B,y_B)$ equals to $k$.

Find locus of all points that its sum of squares of distance between them and point $A(x_A,y_A)$ and $B(x_B,y_B)$ equals to $k$. We know that all the points that their sum of distances between them ...
madfd adfd's user avatar
1 vote
1 answer
97 views

What is the locus of points in the plane $\{v : v \cdot (v-a) = 0\}$ for fixed $a$?

Fix $a$ in $\mathbb R^2$. What is the locus of points $\{v : v \in \mathbb R^2$ and $v \cdot (v-a) = 0\}$? Clearly, $0$ and $a$ are in this set, and no other multiple of $a$ is in the set. Beyond ...
SRobertJames's user avatar
  • 4,450
3 votes
2 answers
119 views

Showing that in a triangle $ABC$, $\angle ABC = 60°$ , the following intersection points, and points $B$, and $C$ lie on the same circle

The problem: In an acute-angled triangle $ABC$, point $H$ is the orthocenter, point $O$ is the center of the circumscribed circle, point $J$ is the center of the inscribed circle, $\angle BAC = 60°$. ...
curioushuman's user avatar
0 votes
0 answers
52 views

Locus of a point Question

The triangle abc is circumscribed in the circle o. The altitudes bb1 and cc1 intersect at h. If bc is fixed and point a moves on the arc bac, determine the locus of point h. my attempt quadrilateral ...
Ahmad Ahmadi's user avatar
3 votes
1 answer
114 views

Proving the locus of $P_k$ is an ellipse.

I was recently working on the following problem functioning within an overlap of complex numbers and coordinate geometry: Let $z$ be a complex number $a + ib$ (where $a > b > 0$), and $α_k$ ($0 &...
Yuv Verdia's user avatar
2 votes
1 answer
78 views

Intersection of perpendicular tangent lines - generalization of directrix?

This is a funny little problem that I came up with. For a differentiable function $f$, define a locus of points $P$ as follows: Let $m$ be an arbitrary tangent line to $f$, and let $n$ be another ...
Steve Dalton's user avatar
3 votes
2 answers
79 views

Locus involving a variable triangle

The base BC of a variable triangle ABC is fixed, and the sum AB+AC is constant. The line DP drawn through the midpoint D of BC parallel to AB meets the line CP at P where CP is a line parallel to the ...
Aarush Saharan's user avatar
2 votes
0 answers
72 views

Finding locus of points $N$ outside a convex quadrilateral $ABCD$ for which $[ABN]- [CDN] = k$

Suppose we are given a positive number $k$ and a quadrilateral $ABCD$ in which $AB$ and $CD$ are not parallel. Find the locus of points N OUTSIDE $ABCD$ for which $[ABN]- [CDN] = k$. Note: $[ABN]$ ...
Jonathan Ramachandran's user avatar
1 vote
1 answer
56 views

Confusion in defining region between regions of complex rays relationships.

The following problem came up while attempting to answer the following: What does $arg(z-z_1) - arg(z-z_2) = \phi$ represents. While, OP themselves tried a different method, one could interpret the ...
Dstarred's user avatar
  • 2,487
1 vote
1 answer
153 views

What does $\arg(z-z_1)-\arg(z-z_2)=\phi$ represents.

What does $\arg(z-z_1)-\arg(z-z_2)=\phi$ represents. where $z$ is point in argand plane My Doubt: I am attaching a Image for the doubt because I don't know How to draw in latex. I'll be grateful if ...
mathophile's user avatar
  • 3,835
0 votes
0 answers
67 views

Shaded area on argand plane if $|\bar z + zi| \leq 2$

The shaded area on the argand plane if $|\bar z + zi| \leq 2$. and find the maximum value for $Arg(z)$ My attempt: Let $$z=x+iy$$ $$|(x-iy)+(x+iy)i| \leq 2$$ Then $$|(x-iy)+(xi-y)|\leq 2$$ so we get $$...
Angelo Mark's user avatar
  • 5,976
2 votes
2 answers
71 views

Area of region covered by the movement of a point

This is a question from BDMO In triangle ABC, AB= 12, BC=20, CA=16. X and Y are two points in segment AB and AC respectively. K is a point in segment XY , such that XK/KY=7/5. If we let X and Y vary ...
Abrar's user avatar
  • 23
0 votes
1 answer
89 views

Find locus of points in first octant giving minimum volume

The problem: Let $\Bbb R_+^3$ be the first coordinate octant in $R^3$. Then for each fixed $\alpha > 0$ describe the geometric location of points $A \in \Bbb R_+^3$, which satisfy the following ...
Maximax67's user avatar
  • 135
-1 votes
1 answer
82 views

Locus of points where ant moves [closed]

An ant is standing at the origin $(0,0).$ It starts moving in the $xy$-plane such that at each step, it moves northwards (in $+y$ direction) or eastwards (in $+x$ direction) with equal probabilities. ...
imposter's user avatar
  • 532
0 votes
1 answer
71 views

Is there any other conic possible?

It is told that a conic is the locus of all the points which satisfies the relation. $$\frac{SP}{PM}=constant$$ Where, SP = distance of the point from fixed point PM = distance of the point from fixed ...
user avatar
0 votes
2 answers
83 views

Is the resultant, the locus of the center of the circle?

If a circle $C$ passing through the point $(4,0)$ touches the circle $x^2 + y^2 + 4x − 6y = 12$ externally at the point $(1,−1)$, then the radius of $C$ is? I have a question here, I have two points ...
Ankit's user avatar
  • 234
1 vote
0 answers
47 views

Solving for Intersections of many lines algebraically

The context: In the analysis of beam structures, one can gain insight from the rigid body kinematics, which show the possible infinitesimal displacements of a given structure. To determine the center ...
Noiv's user avatar
  • 51
0 votes
1 answer
44 views

Locus of points $r$ such that $|r-p| - |r-q| = K$ for some constant $K$ and for fixed points $p$ and $q$

Label the points of a triangle $p$, $q$, $r$ in $\mathbb{R}^2$. Fixing both $p$ and $q$, I need to find the locus of points $r$ such that $$|r-p| - |r-q| = K$$ for some real number $K > 0$ Letting ...
Mark's user avatar
  • 399
0 votes
1 answer
63 views

Problem on Complex Number involving Locus of Ellipse

This is the question: If a complex number z satisfies $\left|z+3\right|+\left|z-3\right|=10$, then the value of $\frac{60\left|z+3\right|}{\left|z+\overline{z}+\frac{50}{3}\right|}$ is? I noticed that ...
Anonymousstriker38596's user avatar

1
2 3 4 5
11