# 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.
1 vote
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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 ...
1 vote
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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 ...
1 vote
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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 ...
1 vote
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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 ...
1 vote
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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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1 vote
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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 ...
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 ...