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Questions tagged [analytic-geometry]

Questions on the use of algebraic techniques for proving geometric facts. Analytic Geometry is a branch of algebra that is used to model geometric objects - points, (straight) lines, and circles being the most basic of these. It is concerned with defining and representing geometrical shapes in a numerical way.

Analytic geometry, also called coordinate geometry, mathematical subject in which algebraic symbolism and methods are used to represent and solve problems in geometry.

The importance of analytic geometry is that it establishes a correspondence between geometric curves and algebraic equations. This correspondence makes it possible to reformulate problems in geometry as equivalent problems in algebra, and vice versa; the methods of either subject can then be used to solve problems in the other.

Analytic geometry was introduced by René Descartes in $1637$ and was of fundamental importance in the development of the calculus by Sir Isaac Newton and G. W. Leibniz in the late $17^{th}$ cent. More recently it has served as the basis for the modern development and exploitation of algebraic geometry.

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What is the polar of an Euclidean unit ball but its center is not in the origin

I know that the polar of an Euclidean unit ball is itself, but I wonder what if its center is not in the origin, like: $$ B=\{(x,y)\in \mathbb R^2 \mid(x-1)^2+y^2\le1\} $$ and the polar of a set is: $$ B^{\circ} = \{ c \in \mathbb{R}^{n} \mid…
user272101
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Analytic geometry question - finding an equation of a line in 3D

I need to find the equation of the line that intersects perpendicularly the line: $$ \frac{x+1}{2}=-y=\frac{z-2}{3} $$ and passes through : $(2,3,1)$. So I know that the line should be of the form: $$ (2,3,1)+t (\ell , m ,n) $$ where: $(\ell , m…
CrazyStatistician
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Finding a point that is a certain distance away from a segment

I have two endpoints $(x_1, y_1)$ and $(x_2,y_2)$ of a line segment. I want to extend the existing segment by a length of $d$ on just one side of the segment. What are the coordinates of the new endpoint? So I have $(x_1,y_1)$, $(x_2,y_2)$. I will…
cgo
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problem related to the slope of a line.

What is the slope of the line given by $\sqrt{x^2+4y^2-4xy+4} + x-2y=1$ . Not getting any start . Only observed we have $(x-2y)^2$ under the root . NOTE: root gets over after 4 so please dont misinterpret. How to proceed any clue would do. Thanks!
Archis Welankar
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Prove that the set of value(s) of $a$ for which the points $P(u,-w)$ and $Q(v,a^2)$ lies on the same side of the line $4x-y+5=0$ is $(-3,3).$

Let $u,v,w$ satisfy the equations $uvw=-6,uv+vw+wu=-5,u+v+w=2,$where $u>v>w$,then prove that the set of value(s) of $a$ for which the points $P(u,-w)$ and $Q(v,a^2)$ lies on the same side of the line $4x-y+5=0$ is $(-3,3).$ I tried to solve…
Vinod Kumar Punia
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A line through the midpoint of the sides $AB$ and $AD$ of a rhombus $ABCD,$whose one diagonal is $3x-4y+5=0$ and one vertex is $A(3,1)$ is

The equation of a line through the midpoint of the sides $AB$ and $AD$ of a rhombus $ABCD,$whose one diagonal is $3x-4y+5=0$ and one vertex is $A(3,1)$ is $ax+by+c=0$.Find the absolute value of $a+b+c$,where $a,b,c$ are integers expressed in lowest…
Vinod Kumar Punia
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Incenter of the triangle formed by the lines whose are $3x+4y=0;5x-12y=0$ and $y-15=0$ is the point $P$ whose coordinates are $(1,8)$

This is a reasoning type question. Statement$(1):$Incenter of the triangle formed by the lines whose are $3x+4y=0;5x-12y=0$ and $y-15=0$ is the point $P$ whose coordinates are $(1,8)$. Statement$(2):$Point $P$ is equidistant from the 3 lines forming…
user1442
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For $a>b>c>0$,the distance between $(1,1)$ and the point of intersection of the lines $ax+by+c=0$ and the $bx+ay+c=0$ is less than $2\sqrt2$

For $a>b>c>0$,the distance between $(1,1)$ and the point of intersection of the lines $ax+by+c=0$ and the $bx+ay+c=0$ is less than $2\sqrt2$,then $(A)a+b-c>0$ $(B)a-b+c<0$ $(C)a-b+c>0$ $(D)a+b-c<0$ I found the point of intersection of lines…
user1442
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Find the least positive value of ordinate of $C$

Let $A(0,2),B$ and $C$ are points on curve $y^2=x+4,$ and such that $\angle CBA=\frac{\pi}{2}.$Then find the least positive value of ordinate of $C$. $y^2=x+4$ is a equation of a parabola whose vertex is $(-4,0)$ and the point $A$ is on the…
Brahmagupta
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What rectangle does this statement convey?

What rectangle is this talking about? The rectangle: $$\{(x,y)\mid 0 \leq x \leq 2 , 0 \leq y \leq 1 \}$$ How does this convey a rectangle? Please help me out. Thank you.
arihant
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Geometric location of points for which the distance to $2x-y+2z-6=0$ is twice the distance to $x+2y-2z+3=0$

how can I find the geometric location on $\mathbb{R}^3$ for which the distance to $2x-y+2z-6=0$ is twice the distance to $x+2y-2z+3=0$ ? The equation I'm used to using to compute the distance from point to plane is $$ d = \frac{| \ Ax + By + Cz + d…
bru1987
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Plane perpendicular to $r \ \{ \frac{x-1}{2} = y - 2 = \frac{z+1}{4}$ that contains $P = (-1,3,-1)$ - possible textbook mistake

I've been working on a problem on a textbook that asks for the following: Find the plane perpendicular to $r \ \{ \frac{x-1}{2} = y - 2 = \frac{z+1}{4}$ that contains $P = (-1,3,-1)$ The options are a) $4x+z-13 = 0$ b) $2x+y+4z-3 = 0$ c)…
bru1987
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Which surface is formed by rotating a hyperbola around its asymptotes?

I don't know even what a type of surface will be. And what equation will be? The equation of hyperbola - $$ xy = l. $$ Now, let's $$ x = x'cos(\varphi ) - y'sin(\varphi ), y = x'sin(\varphi ) + y'cos(\varphi ) \Rightarrow \frac{1}{2}sin(2 \varphi…
John Taylor
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Convert this in to the standard form of an ellipse

$$ \frac{p^2+q^2a^2}{a^2}-\frac{\left(2px+2qa^2y-a^2\right)}{a^2y^2+x^2}=k^2$$ How to put the above equation into this form? $$\frac{\left(X-H\right)^2}{A^2}+\frac{\left(Y-K\right)^2}{B^2}=1$$ The algebra is killing me :) Any help would be…
soupynoodles
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Prove that the four vertices of a quadrilateral,the combined equation of whose sides is $|x^2-y^2|-2(|x+y|+|x-y|)+4=0$,are concyclic.

Prove that the four vertices of a quadrilateral,the combined equation of whose sides is $|x^2-y^2|-2(|x+y|+|x-y|)+4=0$,are concyclic. I need to find the equations of the sides in order to get the vertices.So tried to fetch the individual equations…
Vinod Kumar Punia
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