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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Locus of a moving point, when constraints on an angle and length are given

$APQ$ is a variable triangle; $A$ is fixed, $P$ moves on a fixed line $CD$; if $AP$ meets a fixed line parallel to $CD$ at $R$, and if $PQ=AR$ and if the angle $APQ$ is constant, prove that the locus ...
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finding locus of a point in 3d.

Find the locus of the point which moves so that its distance from the line x=y=z is twice its distance from the plane x + y+z=1. I know the distance of point (x,y,z) from given plane will be mod(x+...
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Locus of a moving point, such that two distances have a common ratio

A, B are two fixed points on a fixed circle; P is a variable point on the circle; Q is a point on BP, such that BQ/AP is constant; find the locus of Q. The only approach I could think of is through ...
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How to find slope of locus?

Given a system of DEs: $$\begin{cases} \dot{x} = g_1(x,y) \\ \dot{y} = g_2(x,y), \end{cases}$$ where $g_1,g_2 \in C^1$ How to show that the slope of $g_1 = 0$ in the neighborhood of the steady state ...
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The locus of points of the form $ae^{i\theta}+be^{i\phi}$

Let $a$ and $b\in\mathbb{R}^{>0}$ be two positive real numbers. What is the locus of points of the form $ae^{i\theta}+be^{i\phi}$ where $\theta$ and $\phi\in[-\pi,\pi)$? Does it have any specific ...
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Modifying the Lemniscate of Bernoulli to have asymmetry along one axis

I'm looking for ideas on how to modify the Lemniscate of Bernoulli to include some asymmetry across one axis. The parametric equations of the lemniscate are $$x(t)=\frac {a\sqrt{2}\cos t }{\sin^2t+...
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locus problem in analytical geometry asking about a constant sum of two tangents to two identical circles yielding an ellipse

You are given two circles: Circle G: $(x-3)^2 + y^2 = 9$ Circle H: $(x+3)^2 + y^2 = 9$ Two lines that are tangents to the circles at point $A$ and $B$ respectively intersect at a point $P$ such ...
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Locus of points in complex plane

I have a problem where I need to find a locus of all points in the complex plane that satisfy $|z-ia|=\lambda|z+ia|$, where $z=x+iy$, and $\lambda>0$. I know I need to get a circle with the radius ...
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Find the locus of point $P$ which lies on a circle.

QUESTION: Consider the circle with radius $1$ and cente at the point $(0,1)$. From this initial position the circle is rolled along the positive x-axis without slipping. Find the locus of the point $P$...
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What is the solution set for point $X$ if $|AX|^2+ |BX|^2=c$?

Let there be distinct points $A$ and $B$ and a given $c\gt0$. What is the solution set for point $X$ if $|AX|^2+ |BX|^2=c$? I started solving this problem as: $\overrightarrow{AB}=\overrightarrow{AX}-...
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finding locus of z using information about z^2

I came across this interesting question,it is as follows: a complex number z satisfies the equation |z^2-9| + |z^2| = 41 and we are required to find the locus of z and maximum value of |z| my ...
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Breakaway point of Root Locus not as expected

I have the following Open Loop Transfer Function: $$H_{ol}s = \frac{k(s+4)}{(s+1)(s+2)(s+3)}$$ To find the breakaway point of the unmatched poles we find the characteristic equation of the closed ...
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Is there an easier way to eliminate a variable from equations?

I was wondering if theres an easier way to do these types of problems that ask to eliminate a variable, since these types of problems are asked in timed tests, and I really can't afford to spend 10 ...
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How can I sketch this complex number set?

Describe and sketch the set of complex numbers satisfying the conditions: (|z|−1)(|z|−2)≤0 and Im(z)>0. I have done the previous part where I had to sketch (|z|−1)(|z|−2)≤0 and Im(z)>0, and that was ...
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A variable parabola touches the $x$-axis and $y$-axis at $A(1,0)$ and $B(0,1)$. Find the locus of its focus.

A variable parabola touches the $x$-axis and $y$-axis at $A(1,0)$ and $B(0,1)$ on the co-ordinate plane respectively. Now, we are required to find the locus of the focus of this variable parabola. ...
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Locus of reflection:Special case

In a triangle ABC, right angled at C, points A and B are fixed. C moves on a circle with AB as diameter.Suppose I is the incentre and the incircle touches the hypotenuse at F. What is the locus of the ...
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Locus of Complex number Collapsing into a single point.

Suppose we have the Locus $$|z-1|=1$$ obviously it is supposed to represent a circle of radius 1 and centered at $(1,0)$. But now if i expand it I get , $$z^{2}+1+2zcos\theta =1$$ which eventually ...
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Locus of a angle point of base of an isoceles triangle with a constant angle while moving other angle point of base on a given straight line.

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. According to me the answer should be two straight lines making ...
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Locus of intersection point of tangents at extremities of chords of $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$

What is the locus of the point of intersection of tangents at the extremities of the chords of the ellipse $$\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1\;\;$$ subtending to a right angle at its center?
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Locus of $\frac{|z-a|}{|z-b|}=c, c<1$ is a circle

I saw similar questions with this one, but I did not find geometrical solution. Therefore, I want to share my solution and also seek other simpler solutions. Problem: I need to prove that the locus ...
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Conics: why is the eccentricity and focal parameter well-defined, and how to show uniqueness of foci and directrices

The focus-directrix definition of a conic says that a (non-circular) conic is a collection of points such that there exists a line $\ell$ called a directrix and a point $F$ called a focus outside the ...
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Determining the locus of z where $|z-(2+i)|=|z|\sin(\pi/4-\arg(z))$

The question is simply having to determine the locus (not the equation, just the conic) of $z\in\mathbb{C}$ where $z$ satisfies $|z-(2+i)|=|z|\sin\left(\frac{\pi}{4}-\arg(z)\right)$. One method is ...
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Locus of segments of length $n$ which form an $n^\circ$ angle with the $x$-axis

So basically, I was wondering what would happen if you take the locus of line segments of length $n$ that make a $n^\circ$ angle with respect to the $x$ axis. I did this, and this is what I got: I ...
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Problem understanding Complex Locus in Cartesian Form

The Question: If A($z1$) and B($z2$) are the two complex numbers satisfying $$ z1: z+(\sqrt{3}+i)t -i = 0$$ $$and$$ $$ z2: \sqrt{3}z+(\sqrt{3}+i)\lambda -\sqrt{3}i=0$$ Where $Arg(z1)=\frac{\pi}{4}$ ...
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Exclusion of certain points of a complex locus

I encountered the following STEP III ($2011,Q8$) problem: The complex numbers z and w are related by $$w=\frac{1+iz}{i+z} \tag{1}$$ Let $z=x+iy$ and $w=u+iv$, where $x$, $y$, $u$ and $v$ ...
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Locus of a point when a line varies

Let $ABC$ an isosceles triangle and $\Delta$ a variable line passing trough $A$. We denote $C’$ the image of $C$ through the reflection across $\Delta$ axis. 1) Find the locus of $C’$ when $\Delta$ ...
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A continuous class of curves (solutions to a polynomial equation) degenerating at some mysterious value! Find that value.

Depending on $a$ -- I am mostly interested in $ -1 \leq a \leq +1 $ -- the solutions of $$ y^{2} -\left(1+x\right)x^{2} = a $$ as a subset of $\mathbb{R}^2$, 1) Looks like a human figure at about $ ...
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Determine complex numbers represented by points common to loci $|z| = 5$ and $|z-5|=|z|$

On a single Argand diagram sketch the loci |z| = 5 and |z-5|=|z|. Hence determine the complex numbers represented by points common to both loci, giving each answer in the exponential form. I know ...
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Finding locus of centre of circle

The locus of the center of the circles such that the point $(2,3)$ is midpoint of the chord $5x +2y =16$ is (A) $2x-5y+11=0$ (B) $2x+5y-11=0$ (C) $2x+5y+11=0$ (D) None of these I ...
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“Sphere Ellipse” locus on surface of sphere

Been attempting again to find a neat equation on a unit radius sphere for Sphere Ellipse locus in 3D conceptually similar to a plane ellipse. Geodesic arc distances between point P, in spherical ...
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Maximum of $\sin(x)$ with increased iterations

My question is about composing sines. When you compose $\sin x$ ($x$ in radians of course) $k$ times, what is the maximum in terms of $k$? For example, composing $2$ sines, as in $\sin(\sin x)$, ...
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Solve for the real scalar value of $k$, if $z =\frac{2}{1+ki} -\frac{i}{k-i}$ and $z$ lies on the line, $y= 2x$

The complex number $z$ is given by $\frac{2}{1+ki} - \frac{i}{k-i}$. If it is given that $z$ lies on the line, $y = 2x$, find the value of the real scalar $k$. So far this is what I have come up ...
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Finding the locus of all the points $C$ so $\angle ACB=\frac{2\pi}{3}$

Let $A$ and $B$ be two different points in the plane. Find the Locus of all the points $C$ so $\angle ACB=\frac{2\pi}{3}$. What I tried to do: Also the center of the circle is in $O(0,0)$. We can ...
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Region created by locus of reflected lines

This may be more a physics topic but I feel like it is a math stack exchange topic. Say you had a circular reflective surface with rays coming in parallel to the surface. They rays reflect with angle ...
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Find locus of Q of an equalateral triangle APQ with A fixed and P moving along a line

ABC is an equilateral triangle with vertex A fixed and B moving in a given straight line. Find the locus of C I drew some figures and realised that locus is a straight line and the part left is to ...
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What is this whole thing called?

What is the figure in this animation called? The black dot is mid point of the green line segment joining the purple dot & the green dot. The green dot rotates 5 times as fast as the purple dot. ...
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What is the locus of mid-point of AB?

A variable line in a plane passes through a fixed point and meets the coordinate axes at points A and B. What is the locus of mid-point of AB? What I did is:- I took a line passing through a fixed ...
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Eliminating $a,b$ from the system $2b+3a=ab$, $x_1=\frac{ab^2}{a^2+b^2}$ and $y_1=\frac{a^2b}{a^2+b^2}$

I am trying to find the locus of foot of perpendicular drawn from the origin to a variable line passing through a fixed point $(2,3)$. My Attempt Let the line be $bx+ay-ab=0$ where $a$ and $b$ are ...
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Square inscribed in a right triangle problem

Let A be a point on a fixed semicircle with diameter BC. MNPQ is a square such that $M \in AB, N \in AC, P \in BC, Q \in BC$. Let D be the intersection of BN and CM and E be the center of the square. ...
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Eliminating a varying parameter for finding locus of foot of perpendicular from $(10,0)$ to any tangent on $x^2+y^2=16$

Locus of foot of perpendicular drawn from a fixed point $(10,0)$ on the $x$-axis to any tangent to the circle $x^2+y^2=16$ is My Attempt Let the tangent be of slope $m$. So equation of tangent is $y=...
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On locus of points such that the product is constant.

I tried to find locus of all points such that the product of distances from two focii is constant. I assumed that the vertex is at (+a,0), and the two focii are (+c,0) and (-c,0). I arrived at the ...
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If |z| = 2, what is the locus of a+bz?

How do I try to find the locus of the complex number $a+bz$ such that $|z|= 2$? I tried putting $z=x+iy $ but it wasn't any help. How do I even start doing this?
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General approach to finding the equation of circle, locus of $\arg(\frac{z-z_2}{z-z_1})=\theta$ where $\theta\ne n\pi, n\in \mathbb{Z}$

Let's say I have an equation of the form $\arg(\frac{z-z_2}{z-z_1})=\theta$ where $\theta\ne n\pi, n\in \mathbb{Z}$ and $\arg(z)$ denotes the principal argument of $z$. I know that the locus of $z$ ...
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What does $ \lvert z-a \rvert = \mathit Re(z)+a $ look like?

What does a loci with the equation look like? $ \lvert z-a \rvert = \mathit Re(z)+a $ This is for the applying complex numbers topic of an advanced HSC maths course. I was asked to describe the loci....
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A Triangle is formed by the lines $X + Y = 0, X - Y = 0$ and $LX + MY = 1$ where $L^2 + M^2= 1$. Find the locus of circumcenter of triangle so formed.

Using basic Geometry I have gotten the coordinates of the circumcenter in terms of $L$ and $M$, I don't have any idea how to proceed further. To obtain the coordinates, solve the three equations ...
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Two distinct tangents drawn from an external point to a cubic

In this question Phillip mentions is his answer The equation $y=x^3+cx$ can have two distinct tangents drawn from an external point only if $c>0$ I tried to prove this: Let the external point ...
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Is there anything like hyperbolic cone?Are the terms elliptic,or circular very precise in analytical geometry for a cone?

What is the locus of a line passing through a fixed point and intersecting a hyperbola.We know that there are things like circular or elliptical cone if the generating line always intersects a circle ...
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Polar Coordinates in a parametric equation

A curve is given parametrically by the equations $x = a\cos^3 t\ ,\ y = a\sin^3t \ $ $ \ 0\le t \le 2\pi$ Where a is a positive constant. Show that the equation of the tangent at the ...
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Locus of points whose distance difference is a constant to three (or more) points

Given two points on a plane, the locus of points with a constant distance difference is a hyperbola. What happens if there are three points on the plane? Concretely, if there are three points A, B, ...
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Question on Locus [closed]

Two straight lines rotate about two fixed points $(-a, 0)$ and $(a, 0)$ in anti-clockwise direction. If they start from their position of coincidence such that one rotates at a rate double of the ...

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