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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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Point on a line with the least distance from another point in $\mathbb{R}^3$

Consider the line $L$ defined by the following parametric equations $$x= 3+2t$$ $$y= 4+t$$ $$z=5-6t$$ Find the point $Q$ on $L$ that is closest to $(4,1,7)$. Note: I do not really remember the formulas, I need help!
Mario De León
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A line is drawn through the point $A(1,2)$ to cut the line $2y=3x-5$ in $P$ and the line $x+y=12$ in $Q$. If $AQ=2AP$, find $P$ and $Q$.

A line is drawn through the point $A(1,2)$ to cut the line $2y=3x-5$ in $P$ and the line $x+y=12$ in $Q$. If $AQ=2AP$, find the coordinates of $P$ and $Q$. I found the lengths of the lines $AQ$ and $AP$ in terms of $x$ and $y$ and used $AQ=2AP$ to…
shredalert
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Prove the median of right triangle is half the length of the hypotenuse

How can I prove that in a right triangle the median which tends to the hypotenuse has length half of that the hypotenuse? I want to show using vector methods or analytic geometry
etet112
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Finding the third vertex of an equilateral triangle

If two vertices of an equilateral triangle are (0,0) and (3,√3) then find the third vertex. The first thing I did was calculated the distance of the given points and tried to make an equation including the third vertex but instead of being…
Ger Wyn
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Shrinking a rectangle enscribed about another, such that it is enscribed by the other

I ran into this problem during my work, and I've been banging my head against it for a bit. Seems like a simple enough algebra problem, but I'm thinking it'll need to be solved numerically. Here's the problem: say there are 2 rectangles, A and B. B…
user295105
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Find the least eccentricity of an ellipse which can rest on the plane.

A perfectly rough plane is inclined at an angle $\alpha$ to the horizontal.Find the least eccentricity of an ellipse which can rest on the plane. Any ideas on how to solve this problem?
user220382
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Find the equation of the image of the plane $x-2y+2z-3=0$ in the plane $x+y+z-1=0$.

Find the equation of the image of the plane $x-2y+2z-3=0$ in the plane $x+y+z-1=0$. I have no idea how to find the image of a plane in another plane. Please help me.
diya
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An equation for all those points that have the same shortest distance to the same straight line in 3D space.

Can you form an equation for a ''pipe'' in 3D space? It means all those points P(x,y,z) that have the same shortest distance for the same straight line l. For example what would the pipe equation be when we know that the line l is x-1=y-1=z-1 and…
user30606
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Analytic Geometry

In our book of analytic geometry we have a title The canonical form of a line. It is the equation of a line passing through a point $p_1 := (x_1 , y_1,z_1)$ and parallel to a vector whose direction ratio is $a:b:c$. Under another title The Symmetric…
Radwa Kamal
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How do I confirm my answer for the orthocenter of a triangle?

I know how to find the orthocenter, but how do do I check to see if I have the right answer? For instance, once I find the centroid of a triangle I can use the centroid formula(a+b+c)/3 to check my answer. Is there anything like that for the…
moh abdi
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Coordinates of two missing vertices of a sqaure

I'm given the coordinates of two vertices of a square: $(5,1)$ and $(8,5)$ and I'm asked to find the other two. I suppose there are two squares which meet these conditions. I can obtain a point $(x,y)$ by solving two equation derived from the…
ZaIROuS
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Let $M$ be a point on the straight line $L\equiv x+y=0$ such that $|AM-BM|$ is minimum,then the area of triangle AMB

Consider two points $A(1,2)$ and $B(3,-1)$.Let $M$ be a point on the straight line $L\equiv x+y=0$ such that $|AM-BM|$ is minimum,then the area of triangle AMB…
user1442
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If the pair of lines $6x^2-\alpha xy-3y^2-24 x+3y+\beta=0$ intersect on $x-$axis,then find the value of $20\alpha-\beta$

If the pair of lines $6x^2-\alpha xy-3y^2-24 x+3y+\beta=0$ intersect on $x-$axis,then find the value of $20\alpha-\beta$. I let the pair of lines cut at $(\lambda,0)$,putting in the equation gives. $6\lambda^2-24\lambda+\beta=0$,but there seems no…
Brahmagupta
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Analytical Geometry

I have Problem in Analytical Geometry Question: The Portion of a Straight line between the axes is bisected at the point (-3,2) Find its Equation My Try I used Equation formula $$(y-y_1)=m(x-x_1)$$ but it requires Slope
Ganesh
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General concepts of lines and planes

I've come across an exercise that asks the reader to state the correct answer. For me, the only one right is the third, but I would like to know why the first one is not correct also: The line $r$ is parallel to the plane $\alpha$, but is not…
bru1987
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