# 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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### 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 ...
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 ...