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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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Can a plane be perpendicular, at the same time, to two other planes if they're not parallel to each other?

I need to find the equation of a plane that passes through a point and is perpendicular to two other planes (equations were given). To my understanding, if two planes are not parallel to each other, there is no plane or line that is, at the same…
Gabriel
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Equation of the line that has $x$ and $y$ intercepts at $a$ and $b$.

Please can anyone help me with proving the following problem: Show that the line that crosses the $X$-axis at $a \neq 0$ and the $Y$-axis at $b \neq 0$ has the equation $$\dfrac{x}a + \dfrac{y}b -1=0$$ I don't have an idea of where I should start…
Samama Fahim
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How can we derive the standard form of the linear equation: $Ax+By+C$?

How can we derive the standard form of the linear equation: $Ax+By+C=0$? What do "$A$", "$B$" and "$C$" in the standard form of the linear equation mean? As in the point-slope form of the linear equation: $y-y_0=m(x-x_0)$, $m$ is the slope, and $x$,…
Samama Fahim
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Slope; A measure of Direction

In my book, the definition of the slope of the straight line is: The slope is a measure of the direction of the line. 1) When the line has no slope, it tells that it is vertical or moving vertically along the $y$ and $y'$ axis. 2) When the slope…
Samama Fahim
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Point-slope Equation

Let suppose that $R(x_1, y_1)$ is a point on the (x, y)plane and a line $L$ with slope $m$ passes through this point. There is a point $S(x_1, y_1)$ on $L$ such that $R$ and $S$ are coincident points. Then the point-slope equation can be given…
Samama Fahim
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Intersection between a plane and a sphere

We have a sphere $(x-1)^2 + (y-1)^2 + (z-1)^2 = 1$ and a point $A = (1;1;-1)$. Find all equations of planes which contain the line $OA$ and intersections with the sphere are circles of radius $\sqrt{3}/3$. Find the centres of the circles. I don't…
Oleg Pavliv
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Difference of two points on a plane

If $P(x_1, y_1)$ and $Q(x_2, y_2)$ are the two points on a plane, then the change in $x$ and $y$ coordinates is denoted by $∆x$ and $∆y$ respectively. Therefore, $x = ∆x = x_2 - x_1$ and $y = ∆y = y_2 - y_1$. The quantities $∆x$ and $∆y$…
Samama Fahim
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$2D$ Line Segment - Triangle Intersection

I've seen similar questions but could not solve my problem with those. My question is how to detect an intersection of a line segment and a triangle on a 2D coordinate system? I don't need the point of intersection, I just quickly need to decide…
Gabor Peto
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Distance Formula Problem

If two vertices of an equilateral triangle are $(1, -1)$ and $(-\sqrt{3}, - \sqrt{3})$, find the coordinates of the third vertex. Step by Step procedure to get the answer. Take $A=(1, -1)$, $B=(-\sqrt{3}, - \sqrt{3})$. Let the third vertex be $C=(x,…
Club Player
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Area/length of this peculiar lemniscate

I discovered this curve that looks like a lemniscate. It’s defined as the set of points $P$ in ${\Bbb R^2}$ that satisfies the equation $$\frac{1}{d_{-1}(P)}+\frac{1}{d_{1}(P)}=2,$$where $d_{a}(P)$ is the distance from $P$ to the point $(a,0)$. It…
Ludvig Lindström
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Plugging in a point not on a plane, into the plane's equation

Say a plane P has a given equation ax+by+cz=d. Given a point $(x_0, y_0, z_0)$ that is not included in P. When $(x_0, y_0, z_0)$ is plugged into $f(x,y,z)=ax+by+cz-d$ and it outputs some nonzero $e$. What does $e$ represent? What information does…
Mr.Guy
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How do you know that $y= \sqrt{1-x^2}$ has the shape of a semicircumference?

A way to explain this is probably that when you square them and then reagroup terms you'll end up having that $y^2+x^2 =1$ and that forms and circumference, but since $y$ can't be negative it becomes a semicircumference. In other words, what I am…
Vmimi
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Is it possible that the one boat is behind the other one?

We have the equations $g_1: (4,14,-5)+t(2,-3,0)$ and $g_2:(11,9,-15)+t(-2,-3,2)$ on which two boats move at time $t$. We have the point $P(12,11.5,0)$. Is it possible that the one boat is behind the other one when we look from the point $P$? Do we…
Mary Star
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Distance between cone and point

How can I find minimum distance between cone and a point ? Cone properties : position - $(0,0,z)$ radius - $R$ height - $h$ Point properties: position - $(0,0,z_1)$
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Intersection of cylinder and plane

If a cylinder is $x^2+8y^2=1$ and a plane is $x+y+3z=0$, what's the form of the intersection? I thought of substituting the $y$ variable from the plane's equation in the cylinder's equation.
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