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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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Least squares intersection of three circles

Given is a triangle in the plane, with the coordinates of all three vertices known. I need to determine the location of a point $X$, for which the distances to all three triangle vertices are given (with some error). Therefore, $X$ is located at the…
koletenbert
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To find the x and y-intercepts of the line $ax+by+c=0$

Please check if I've solved the problem in the correct way: The problem is as follows: Find the points at which the line $ax+by+c=0$ crosses the x and y-axes. (Assume that $a \neq 0$ and $b \neq 0$. My solution: We have to find the x and…
Samama Fahim
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Prove that two lines are perpendicular to each other if and only if their slopes are negative reciprocals of each other

Prove that two lines are perpendicular to each other if and only if their slopes are negative reciprocals of each other Proof: Let $l_1$ and $l_2$ be arbitrary lines. $(\rightarrow)$ Suppose that $l_1$ and $l_2$ are perpendicular. Then the angle…
Iyeeke
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Algebraically determine if lines intersect

I have a programming problem that involves determining if any 4 line segments intersect. (I am testing to see if four points [in a specific order] comprise a quadrilateral). Mathematically speaking, using the four "endpoints," what is the easiest…
Sir Winford
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The triangle that has the longest possible smallest side of a triangle inscribed in a unit square is equilateral

The points $A$, $B$ and $C$ lie on the sides of a square of side $1$ cm and no two points lie on the same side. Show that the length of at least one side of the triangle $ABC$ must be less than or equal to $(√6−√2)$ cm. The given result can easily…
twentyyears
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To find the intersection of line and a plane.

To find the intersection of line and a plane. Line: $x=y-1=5z$ and Plane : $4x-y+3z=17$. Let $x=y-1=5z =t$, then $x=t, y=t+1$ and $z = t/5$. Thus puttinf this into the equation of plane we have $ t =5.$ Thus the line intersects the plane at $(5,…
User8976
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Find the equation of a plane given point, parallel line and angle between plane and line

Find the equation of a plane that crosses point $P(-1,2,1)$, that is parallel to the line $p: x=0,y=-z$ and its angle with line $q: x=y, z=0$ is $\frac{\pi}{4}$. First lets write those two lines in canonical form. Lets observe normal vectors of…
Filip Vlaisavljević
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How can I find the intersecting curve? $x=6\cos (t+\frac\pi 3).y=3\sin t,z=u \text{ and } 0\leq t\leq 2\pi .u\in R$

Question: Parametric equation of surface S is given below: $$x=6\cos (t+\frac\pi 3).y=3\sin t,z=u \text{ and } 0\leq t\leq 2\pi .u\in R$$ Find the curve which is intersecting with surface S's plane: $y={x\over\sqrt 3}$ and explain what type of…
Matthers
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Parabola transformation

Find the real affine change of coordinates that maps the parabola in the $xy$-plane to the parabola in the $uv$-plane $$4x^2 + 4xy + y^2 - y + 1 = 0$$ $$4u^2 + v = 0$$ My attempt: Since there is an $xy$ term, we know that there is a rotation. Thus…
Tarang Saluja
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Finding the equation of a hyperbola

Find the equation of the hyperbola with center on $2y+x-1=0$, with an asymptote $y+2x-5=0$, and a focus $(1,0)$. Can anyone help me out with this problem?
S bila
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Circles of radius $2$ passing through origin with centers on $x=1$

There are two circles of radius $2$ that have centers on the line $x=1$ and pass through the origin. Find their equations. Please explain to me what the problem is really saying.
Samama Fahim
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Locus of the equation

One way to describe a set of points in the plane is by an equation or inequality in two variables, say $x$ and $y$. A solution of an equation in $x$ and $y$ is point $(x_0, y_0)$ in the plane for which the equation is true. My questions: How do…
Samama Fahim
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Investigating the geometric patterns of $x^k+y^k=r^k$ for $k \in N$

I was investigating the graphs of equations of the form $x^k+y^k=r^k$. I am not sure how to ask this so I will try to simplify the problem first. For simplicity sake lets let $r=2$, now For $k=1$, I get a line. $x+y=2$. For $k=2$, I get a circle…
Deathvulken
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How to differentiate between external and internal angle bisectors of a triangle?

I came up with this question when I was trying to figure out the coordinates of the incenter of a triangle with equations: $4x-3y=0$, $3x-4y+12=0$, $3x+4y+2=0$. I assumed the coordinates of the incenter to be $(h,k)$ and equated the perpendicular…
asks281
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proving or disproving that two tangent lines are parallel to a curve

Im trying to prove or disprove that given the function, $f(x)=0.5\sqrt{1-x^{2}}$, There are two different tangent lines to $f(x)$ so they are parallel. I tried to derivative but with no success.
user64370
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