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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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Normal vector to surface

This is a very noob question, but can someone please give me an example of finding the normal vector to a surface (if this is the word in English) which is defined by three points in it. I know that there are two ways - the first to find the…
Pretty_lemon
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Area of triangle given 2 coordinates and a line equation

Problem: The vertices of the base of an isosceles triangle are $(-1,-2)$ and $(1,4)$. If the third vertex lies on the line $4x + 3y = 12$, find the area of the triangle. Attempt 1 : Convert $4x + 3y = 12$ to point slope form which is $(y-0) =…
Jayce
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Finding the set of points (a,b) in an xy-plane for which |a|+|b| = 5

I came across this question in an SAT Math Level 2 Subject Test book and the answer confuses me: Question: Which of the following describes the set of points (a,b) for which |a|+|b| = 5 in the xy-plane? Answer Choices: A) A circle with radius 5 B) A…
MapleAce
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Let AB be a fixed line segment. Let P be a moving point such that angle APB is equal to a constant acute angle. Then point P moves along which curve?

Answer is: The boundary of union of two identical intersecting circles with centers outside the common region.
Priyanka
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How many steps do I need?

The start point is $(0,0,0)$. The "step vector" is $(2,3,6)$. I can variate this, so it can be $(3,6,2)$ and so on, and it can be negative too, like $(6,2-,3)$. I can't step over $(99,99,19)$ and I can't step under $(0,0,0)$. What is the minimum…
valaki plda
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Coordinate geometry prove for equilateral triangle area

Equilateral triangle OAB is drawn such that vertex O is in the origin and vertex B passes though line $ax+by+c=0$. So that $ OAB\angle =OBA\angle = 60^\circ $. Prove the area of triangle is $\frac {c^2} {(\sqrt3) (a^2+b^2)}$ I tried to solve this…
emil
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Geometric significace of oblique parabola parameters

What is the geometric significance of constants $ (h,k,m) $ in this oblique parabola equation? $$ y= m x \pm \sqrt{m x h + k^2}$$ Graph made with $ (h,k,m)=( 2,-1,0.5)$
Narasimham
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Find the slope of two linear functions knowing only the intercept of each function and the properties at the intercept of the functions

After some initial confusing on my side I decided to edit the post (and the title). The reason for this is (as many people pointed out) that my original question was (I) confusing and (II) the equations had no solution. So in order to still solve…
PAS
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Is it possible to find intersection of circle and triangle?

Is it possible to find a the exact points a circle will intersect the legs / hypotenuse of a Right 45° triangle? In the picture above it would be the points that are green which are unknown, but the circle radius and location are known as well as…
Robbie
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Proof that a quadratic graphs as parallel lines.

I define parallel lines in the Euclidean plane as two lines with a constant separation between points closest to each other on opposite lines. We might also, just say that lines are parallel when they share proportional direction vectors. To prove…
Narlin
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Analytic of a line

Given points $A(-4,-4), B (-8,-2)$ and $C(x,0)$, what is $x$ If $AC-CB$ has its greatest value If $AC+CB$ has its smallest value I know that if $A$ and $B$ and $C$ are collinear $AC-BC$ has its greatest value so $x$ is $-12$ and $AC+BC$ also has…
Marva Jami
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Equiletral triangle problem

A vertex of an equiletral triangle is $A(2,3)$and the equation of opposite side is $x+y+2=0$ Find the equations of other two sides?
Marva Jami
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Two touching circles, common tangent, proof

Two circles A and B are touching and have a common tangent which meets A at M and B at N. Let MP be a diameter of A and let the tangent from P to B touch it at Q Show that MP=PQ I've drawn it out and a few different lines through the shape to try to…
user3709
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distance between two line as least square problem

Given that $P=(x,x,x)$ and $Q=(y,3y,-1)$ are two lines in $\mathbb{R}^3$ then I need to express in matrix form $\|AX-b\|^2$ to find the distance between $P$ and $Q$. I need to find two points on this line which are closest to each other. I tried…
Myshkin
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Analytic analyse of triangle

If the area of triangle ABC with the vertices $A(3,1)$, $B(-1,-2)$ and $C(m,1)$ is five square units find the possible values of m? I found the answer to be 19/3 but actual answer is 19/3 and -1/3 I solved the question using area of a triangle…
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