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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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 ...
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Is locus changed if we shift coordinate axes [closed]
I mean if a locus a parabola or circle or anything else, will it be same shape parabola or circle or anything else if we shift the origin of the coordinate system.
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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 ...
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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 ...
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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, ...
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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&...
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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 ...
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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 ...
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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 ...
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$z_1, z_2, z_3\in\mathbb C$ 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}-...
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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. ...
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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 ...
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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 ...
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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, $\angle BAC = 60°$. ...
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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 ...
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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 &...
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How to Prove the Apollonius Circle?
Suppose we are given two points, say $A$ and $B$. Point $P$ is moving such that $$\frac{PA}{PB} = r \ne1,$$
where $r$ is constant. It is a well known result that the locus of $P$ is circle. This is ...
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Locus of intersection of normals of hyperbola
$\text{In general four normals can be drawn to the }$ $\text{rectangular hyperbola $xy = c^2$ from }$
$\text{any point P(h,k)}$
$\text{If these normals intersect the curve at }$ $A(ct_{1}, c/t_{1}), B(...
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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 ...
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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 ...
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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: $[ABN]$ ...
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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 ...
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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 ...
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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 $$...
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Area of region covered by the movement of a point
This is a question from BDMO
In triangle ABC, AB= 12, BC=20, CA=16. X and Y
are two points in segment AB and AC
respectively. K is a point in segment XY , such
that XK/KY=7/5. If we let X and Y vary ...
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Find locus of points in first octant giving minimum volume
The problem: Let $\Bbb R_+^3$ be the first coordinate octant in $R^3$. Then for each fixed $\alpha > 0$ describe the geometric location of points $A \in \Bbb R_+^3$, which satisfy the following ...
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Locus of points where ant moves [closed]
An ant is standing at the origin $(0,0).$ It starts moving in the $xy$-plane such that at each step, it moves northwards (in $+y$ direction) or eastwards (in $+x$ direction) with equal probabilities. ...
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Is there any other conic possible?
It is told that a conic is the locus of all the points which satisfies the relation.
$$\frac{SP}{PM}=constant$$
Where,
SP = distance of the point from fixed point
PM = distance of the point from fixed ...
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Is the resultant, the locus of the center of the circle?
If a circle $C$ passing through the point $(4,0)$ touches the circle $x^2 + y^2 + 4x − 6y = 12$ externally at the point $(1,−1)$, then the radius of $C$ is?
I have a question here, I have two points ...
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Solving for Intersections of many lines algebraically
The context:
In the analysis of beam structures, one can gain insight from the rigid body kinematics, which show the possible infinitesimal displacements of a given structure.
To determine the center ...
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Locus of points $r$ such that $|r-p| - |r-q| = K$ for some constant $K$ and for fixed points $p$ and $q$
Label the points of a triangle $p$, $q$, $r$ in $\mathbb{R}^2$.
Fixing both $p$ and $q$, I need to find the locus of points $r$ such that $$|r-p| - |r-q| = K$$
for some real number $K > 0$
Letting ...
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Problem on Complex Number involving Locus of Ellipse
This is the question:
If a complex number z satisfies $\left|z+3\right|+\left|z-3\right|=10$, then the value of $\frac{60\left|z+3\right|}{\left|z+\overline{z}+\frac{50}{3}\right|}$ is?
I noticed that ...
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Find a complex number that satisfies both $|z-3+i|=3$ and $|z+2-i|=|z-3+2i|$.
What I did:
Let $z=x+yi$, Then:
$$|(x-3)+(y+1)i|=3$$
$$|(x+2)+(y-1)i|=|(x-3)+(y+2)i|$$
Since $|x+iy|=\sqrt{x^2+y^2}$:
$$(x-3)^2+(y+1)^2=9$$
$$y=\frac{5x-4}{3}$$ Hence:
$$(x-3)^2+\bigg(\frac{5x-1}{3}\...
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Is negative $x$-axis not the locus of centers of circles touching the $y$-axis and circle $x^2+y^2-2x=0$ externally?
Find the locus of the centre of circles touching the y-axis and circle $x^2+y^2-2x=0$ externally.
My Attempt:
Circle touching y-axis is $(x-h)^2+(y-k)^2=h^2$.
If this is to touch the given circle ...
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Geometry - locus of points
☕ Hello
I was doing an example when this question came to my mind.
The question I was dealing with: two dots (A and B) exist on the paper. Draw a circle with a radius of X which passes through the ...
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Probability of center of rectangle being inside the circle which is formed by taking 2 random points inside the rectangle as the diameter.
What is the probability of center of rectangle being inside the circle which is formed by taking 2 random points inside the rectangle as the diameter?
Suppose that the probability is x/y then x and y ...
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Find the locus of points |z-1|= -Im(z).
If I wish to find the locus of complex points satisfying $ |z-1|= -\text{Im}(z)$, then would I be right in supposing it represents the half-circle
$(x-1)^2 + (y-1/2)^2 = 1/4, y \leq 0$?
My work ...
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Locus of points traced by the Chebyshev lambda linkage
I was looking around on the interwebs when I found a GIF of the locus of points traced by this mechanism called the Chebyshev lambda linkage, and I was curious if there is an equation that describes ...
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A rod sliding on two mutually perpendicular axes
A rod of length $l$ slides on two mutually perpendicular axes. At the ends of the rod, two lines making angles $60^\circ$ and $30^\circ$ with the rod are drawn then prove that the locus of point of ...
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Find the locus of the mid-points of the chord of the ellipse $b^2 x^2 + a^2 y^2 = a^2 b^2$ which subtends a right angle at the origin
In a book, I found the solution is $(a^2+b^2)(b^2x^2+a^2y^2)^2=a^2b^2(b^4x^2+a^4y^2)$. Here is my solution:
Let $P$ and $Q$ the ends of a ellipse chord $b^2 x^2 + a^2 y^2 = a^2 b^2$ which subtends a ...
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STEP 1 1999 Question 2, Help with justifying last step
Question:
A point moves in the x-y plane so that the sum of the squares of its distances from the three
fixed points $(x_1, y_1)$, $(x_2, y_2)$, and $(x_3, y_3)$ is always $a^2$
. Find the equation of ...
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The locus of the intersection point of two perpendicular tangents to the curve $xy^2=1$.
Find the locus of the intersection point of two perpendicular tangents to the curve $xy^2=1$.
I find out that the tangents to this curve with slope $m$ has this general form:
$y = mx+(-2m)^{\frac{1}{3}...
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Help with finding this locus of points
Let $(a, b)$ be a fixed point, and $(x, y)$ a variable point, on the curve $y = f(x)$, $(x \geq a, f
′
(x) \geq 0).$
The curve divides the rectangle with vertices $(a, b),(a, y),(x, y)$ and ($x, b$) ...
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Prove that the locus is tangent to the circle
$O,A$ and $B$ are arbitrary points on the plane. Point $C$ moves on the circle with center $O$ and radius $OB$.
Construct a circle with center $C$ and externally tangent to the circle with center $A$ ...
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How to find the locus of midpoint for rectangular hyperbola's chord
given a straight line $y-6=m(x-4)$ passing through $(4,6)$ and meets $xy=4$ at two points, $P$ and $Q$. The midpoint of $P$ and $Q$ is $(\cfrac{2m-3}{m}, 3-2m)$. Find the locus of the midpoint of all ...
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Root locus, deriving the behavior at infinity
I'm reading Control Systems Engineering of Nise, Appendice M.1, Derivation of the Behaviour of the Root Locus at Infinity (Kuo, 1987).
At one point, we have;
$$
f(s) = (1+\frac{b1-a1}{s})^{1/n}
$$
is ...
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Find the locus of a point, sum of whose distances from $(c, 0)$ and $(-c, 0)$ is constant and greater than $2c$
Find the locus of a point, sum of whose distances from $(c, 0)$ and $(-c, 0)$ is constant and greater than $2c$.
Try
Let the point be $(h, k)$. Then by given condition
$\sqrt {(h-c) ^2+k^2} + \sqrt {(...
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Does this locus have a name?
This comes from variational analysis and in particular from the definition of the tangent cone of a set.
Say we have a circle centered at $(0, 1/\tau)$ with radius $1/\tau$, $\tau > 0$. All of ...
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What is the centre of the fixed circle on which the projection of point $A(1; 0; 3)$ on plane $(P_m)\colon(m - 1)x + (2 - m)y - mz + (2m - 1)=0$ runs?
In a three-dimensional Cartesian coordinate system, consider parameter $m$ and plane $$(P_m) \colon (m - 1)x + (2 - m)y - mz + (2m - 1) = 0$$
What are the coordinates of the centre of the fixed ...
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The locus of a line in 3D passing intersecting two other lines
A line through any point on the curve $x^2-y^2=1 , z=0$ intersects two lines $y=x, z=1$ and $y=-x, z=-1$. Required is the locus of the line . Just to clarify(as I have seen people question), the above ...
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