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Questions tagged [locus]

For problems that involve a specific set of locations of points.

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A locus question [closed]

A is a give 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. The answer should be a line ,but i do not know how to prove it . ...
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3answers
52 views

Equilateral $\triangle ABC$ has $A$ fixed and B moving in a given straight line. Find the locus of $C$.

$\triangle ABC$ is an equilateral triangle with vertex $A$ fixed and $B$ moving in a given straight line. Find the locus of $C$. I think the locus of $C$ should also be a line. What I have done: ...
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0answers
35 views

A plane passes through a fixed point (p,q,r) and cut the axes in A, B, C. Show that the locus of the centre of the sphere OABC is p/x + q/y + r/z = 0 [closed]

I know how to find the equation of a sphere when four points, through which it passes is given. But here I'm unable to find coordinates of points $A, B, C$ in terms of $p, q, r.$
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$P$ is at constant distance $2$ from point $(3,5)$. Find the equation of the locus of $P$.

The question states: P is at a constant distance of two units from the point (3,5). Find the cartesian equation of the locus of the set of points P in each case. To solve this I drew it out, but ...
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1answer
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Locus circle and equilateral triangle question

An equilateral triangle of side 25cm circumscribes a circle. Find the radius of the circle. I drew it out, and tried to create a right angled triangle and perform Pythagoras but it was not right, do ...
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1answer
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Loci circle question

Im not sure how to answer this. PS AND PT are two tangents draw from a point P to a circle whose centre is O. Join PO and prove that PT = PS. I drew the diagram out and so I would end up with two ...
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1answer
25 views

Deducing the locus of a point of intersection of two lines.

A straight line $L$ through origin meets $x+y=1$...$(1)$ at $P$ and $x+y=3$...$(2)$ at $Q$. Through $P$ and $Q$ two lines $L_1$ and $L_2$ are drawn which are parallel to $2x-y=5$...$(3)$ and $3x-y=5$.....
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5answers
41 views

Locus of a complex number $z$ when locus of $z^2$ is known

If $|z^2 -1| = |z|^2 +1$ then $z$ lies on a: a) circle. b) parabola. c) ellipse. d) straight line. My attempt: Since $|z|^2 +1$ is some constant value hence the locus of $z^2$ is a circle ...
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1answer
11 views

Reconciling two solutions to locus problem involving area of triangle and equations of the sides.

I was solving the following problem where I was unable to reach the same conclusion using two methods. The problem : The area of a triangle formed by the intersection of the line parallel to X-axis ...
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1answer
14 views

Reason behind alternate answers to a locus problem

The problem :Locus of mid point of a segment (I don't have the required reputation to ask my question regarding an answer here in the comments) A variable line, drawn through the point of ...
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1answer
33 views

Find the locus of point $E'$.

$E$ is a point of side $BC$ of $\square ABCD$. $DE \cap AB = \{D'\}$, $AE \cap CD= \{A'\}$ and $A'B \cap C'D = \{E'\}$. Find the locus of point $E'$. This question was in a competition in 1999. ...
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1answer
27 views

Locus of the circles touching another circle

Find the locus of centre of all circles which are of given radius and touch a given circle. I can make out that it is a circle but I am unable to prove it.
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4answers
35 views

Describe the set whose points satisfy the relation $|{z-1\over z+1}|=1$

Describe the set whose points satisfy the relations $$|{z-1\over z+1}|=1$$ for any $z=a+ib\in\mathbb{C}$. Solution: $$ \\ 1=|{z-1\over z+1}|=|{(z-1)(z^*+1)\over (z+1)(z^*+1)}| \Rightarrow \\|z+1|^2=|...
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1answer
51 views

Locus of orthocenter of triangle inscribed in ellipse

While messing around with ellipses in Geogebra, I found the following interesting result: Let $\alpha$ be an ellipse. Let $AB$ be a fixed chord, and let $P$ be a point that moves freely on $\alpha$. ...
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1answer
38 views

Find all points M such that area(MBC)=area(ABC)

Let ABC a triangle, find the set $\Gamma$ of points M so that $Area(\triangle MBC)=Area(\triangle ABC)$ without using dot product. I treat $Area(\triangle ABC)$ as constant $ \delta$ and define $h$ ...
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2answers
68 views

How to find the minimum distance from origin to locus of P?

A straight line through $A(6,8)$ meets the curve$2x^2+y^2=2$ at $B$ and $C$. $P$ is such a point on $BC$ that the distances $AB, AP, AC$ are in Harmonic Progression. Find minimum distance from origin ...
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1answer
92 views

What is the locus defined by these equations?

I would like to know what is the locus of $x \in \Bbb R_+^n$ ($n=2$ would already be fine) defined by $\sum a_i \cdot x_i$ s.t. $a_i+\epsilon \geq 0$, $\epsilon \in \Bbb R$. I know that if $\...
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0answers
133 views

Proving electrostatic analogy for root locus

My teacher told us that there is a provable mathematical analogy between root locus and the lines of force generated by electric charges, where every pole can be associated to a positive charge and ...
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1answer
40 views

Understanding a proof concerning the loci of zeros of a polynomial curve

I am trying to understand following proof. One definition the author uses: The locus of zeros of a function $f(z, K)$ with respect to $K > 0$ is the set of all points $z$ such that for some $K_0$, ...
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1answer
37 views

Given $n$ points, what is the locus of points $X$ such that the sum of the squares of the distances from $X$ to each point is a given constant?

Given $n$ points $P_{1},P_{2},\dots P_{n}$ and a real number $c,$ find the locus of points $X$ such that $$\sum_{i=1}^{n}XP_{1}^{2}=c.$$ Actually, I'm also interested in a more general case: Given $n$...
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A curve through two vertices of a triangle, whose tangent lines bisect the area of that triangle

Say we have a triangle $ABC$. I want to find a curve $\gamma:[0,1]\to\mathbb{R}^2$ such that $\gamma(0)=A$, $\gamma(1)=B$ and for all $t\in(0,1)$ the tangent line at $\gamma(t)$ divides $\triangle ABC$...
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Linking complex loci to vectors

Find the locus of $z$ such that $$\arg\left( \frac{ z^2 - 1}{ z^2 + 1} \right) = 0~, \qquad z \neq \pm i$$ I am able to solve this by substituting $z$ as $x+iy$ and proceeding algebraically. My ...
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1answer
48 views

Locus of points, analytic geometry

From point B(0,b) and B'(0,-b) two lines perpendicular to Y-axis are drawn, as BN and B'N' respectively - towards the right. The multiplication of the lengths of those lines is $4a^2$. Find the set of ...
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1answer
57 views

Locus of points with parabola

On parabola $y^2=2px$ at point $A$, a line $L_1$ passes that is tangent to the parabola and cuts $x$ axis at point $B$. From $A$, a line $L_2$ passes that is perpendicular to $x$ axis and cuts the ...
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2answers
33 views

Locus/set of points - circle

From point A on the circle $x^2+y^2=R^2$ two lines pass; one that is perpendicular to X-Axis and passes X-Axis at point B, and a second one that is perpendicular to Y-Axis and passes through the ...
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0answers
81 views

Prove the locus of intersection of 2 circles on 2 sides of triangle as chords is hyperbola.

Question Prove that the locus of the intersection of two equal circles which are described on two sides $EF$ and $EG$ of a triangle as chords is a rectangular hyperbola whose center is the midpoint ...
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1answer
68 views

Locus of all midpoints. [closed]

Two circles intersect at $A$ and $B$, $PAQ$ is a straight line through $A$ meeting the circles at $P$ and $Q$. Find a locus of a midpoint $PQ$. Please give only hints and not the solution. This is a ...
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How many points are required to fix a given curve from a locus

Say I have a locus on the $x,y$ plane defined by the parametric implicit function $$F(x, y, p_1, p_2, ..., p_n)=0$$ where $p_1, p_2, .., p_n$ are parameters. I now want to constrain $F$ to pass ...
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2answers
134 views

Locus of Midpoints of chords in a circle.

This question is a Conics/Locus problem: The circle $x^2+y^2=25$ cuts the y axis above the x axis at A. Find the locus of the midpoints of all chords of this circle that have A as one endpoint. I’ve ...
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1answer
30 views

How do you know what part of the semicircle it is when $\arg\dfrac{z-2}{z+2} = \dfrac\pi2$ [closed]

I'm not quite sure how to determine whether the semicircle is below or above the x-axis, because in $\arg\dfrac{z-2}{z+2} = \dfrac\pi2$, it lay above the x-axis with a locus of $(4-x^2)^{1/2}$.
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1answer
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Length of the arc of locus of a complex number

Let z be a complex number satisfying $$\arg\bigg(\frac{z^3-1}{z^3+1}\bigg) = \frac{\pi}{2}$$ on the Argand plane. Then, length of the arc of the locus of z for which $Re(z)>0$ and $Img(z)<0$ (...
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4answers
117 views

Determine all points $P$ on $\triangle ABC$ so that $|\triangle PAB| = |\triangle PBC|= |\triangle PAC|$.

Determine all points $P$ on $\triangle ABC$ so that $|\triangle PAB| = |\triangle PBC| = |\triangle PAC|$. Here, $|\triangle XYZ|$ denotes the area of $\triangle XYZ$. I've tried drawing it up, and $...
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2answers
83 views

Find a locus of points

Given a triangle $ABC$, $A'$ and $B'$ halves $BC$ and $AC$. We have a variable point on a line $AB$. Parallel to $AA'$ and $BB'$ through $P$ cuts $AC$ in $E$ and $BC$ in $F$. Now line $EF$ cuts $AA'$ ...
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1answer
46 views

A Locus problem

Two fixed points $A(p, q)$ and $B(r, s)$ and a fixed line $L: ax + by + c = 0$ are given. A variable point $P$ moves on the plane such that $AP$ intersects $L$ at $C$ and $BP$ intersects $L$ at $D$. ...
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0answers
34 views

Drawing a locus

Consider two given circles of radii $r_1$ and $r_2$ with centres $C_1$ and $C_2$. A point $P$ is such that $\frac{r_1}{r_2} = \frac{PC_1}{PC_2}$. I wanted to know how the locus of $P$ would look like. ...
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1answer
40 views

Complex Loci with Arguments

I need to find the locus of points (on an Argand diagram) such that: (i) $\arg(z-(-1-4i)) + \arg(z-(5+8i)) =0$ (ii) $\arg(z-(-1-4i)) + \arg(z-(5+8i)) = \pi/2$ I could not see a way to solve these ...
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1answer
30 views

Area of region bounded by locus of a point P

The area of the region bounded by the locus of point P satisfying d(P,A)=4, where A is (1,2) is _______ . Where we define the distance between two points P(x,y) and Q(a,b) as $$d(P,Q)=max(|a-x|,|b-y|)...
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25 views

Locus problem defined through distance

Question If the distance of any point $(x,y)$ from the origin is defined as $$d(x,y)=max(|x|,|y|)$$, $d(x,y)=a$, non zero constant,then the locus is___. My attempt In question the equation for d is ...
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0answers
32 views

Complex inequalities Cartesian equation

So I got the following problem and I wanted to solve and find the region using the Cartesian equations. However when I found the Cartesian equations and graphed the regions the answer is incorrect. ...
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1answer
68 views

Complex number inequalities

I got the following question below : I tried and got this as the region , is this correct ?
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2answers
398 views

Locus problem for vertex of equilateral triangle

Question Given an equilateral triangle $PQR$ where $P(1,3)$ is a fixed point and $Q$ is a moving point on the line $x=3.$ Find the locus of $R.$ My attempt Take $Q$ as $(3,p)$ and $R(h, k).$ Then ...
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2answers
59 views

The tangent at $P$ to $y = x^2 - x^3$ meets the curve again at $Q$. Show that locus of midpoint of $PQ$ is $y=1-9x+28x^2-28x^3$

The tangent at a variable point $P$ of the curve $y = x^2 - x^3$ meets it again at $Q$. Show that the locus of the middle point of $PQ$ is $$y = 1 - 9x + 28x^2 - 28x^3$$ My approach $$y^\prime=2x-...
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2answers
86 views

What will be the locus of the intersection point of two tangents to $x^{2/3}+y^{2/3}=a^{2/3}$ that are perpendicular to each other?

If $(a\cos^3 m, a\sin^3 m)$ is a point on the curve $x^{\frac{2}{3}} + y^{\frac{2}{3}} = a^{\frac{2}{3}}$, then find the locus of the intersection point of two tangents that are perpendicular to each ...
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1answer
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Complex Analysis: Locus of $z$ satisfying $|z-4|=4|z|$

Just want to know if I'm on the right track here. let $z=iy$ $|z-4| = 4|z|$ $\Rightarrow \sqrt{(x-4)^2 +y^2} = 4\sqrt{x^2 + y^2}$ $\Rightarrow (x-4)^2 +y^2 = 16x^2 + 16y^2$ $\Rightarrow x^2 -8x -16 -...
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4answers
43 views

Complex Analysis: Locus of $z$ satisfying $|z+1| = |z-1|$

Suppose I want to find the locus of the point $z$ satisfying $|z+1| = |z-1|$ Let $z = x+iy$ $\Rightarrow \sqrt{(x+1)^2 + y^2} = \sqrt{(x-1)^2 + y^2}$ $\Rightarrow (x+1)^2 = (x-1)^2$ $\Rightarrow x+...
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1answer
51 views

elementary locus problem

A point $P(x,y)$ moves in such a way that its distance from the point $A(3,1)$ is always three times its distance from the straight line $x=-1.$ My attempt is $${\sqrt {(x-3)^2 +(y-1)^2}} = 3{\sqrt {(...
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1answer
73 views

Find all points within the locus of a line segment

First time poster so please be gentle! I really should remember this solution from high school maths, but I'm afraid that was a very long time ago :( I am looking for a formula to determine if a ...
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3answers
38 views

How to algebraically determine locus of points of $|z|=|z-2|$? [closed]

How would I algebraically determine the locus of points in the $z$-plane that satisfy the equation $|z|=|z-2|$?
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2answers
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Locus in the case of an ellipse.

Given that $ S $ is the focus on the positive x-axis of the equation$ \frac{x^{2}}{25} + \frac{y^{2}}{9} =1$. Let $P=(5 \cos{t}, 3 \sin {t})$ on the ellipse, $SP$ is produced to $Q$ so that $PQ = 2PS$....
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1answer
34 views

Finding direction using sun

In this vid why does the shadow trace a straight line instead of some curve ? Intuitively it makes sense because at morning the shadow falls to west and at evening the shadow falls to east. He is ...