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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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Find equation of hyperbola?

The hyperbola has center $(0,0)$, and goes through the points $(3,1)$ and $(9,5)$ and the coordinate axes are the symmetry axes. The correct answer is $x^2 - 3y^2=6$.
Jake
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General Coordinate Geometry Problem - How to deal with lines parallel to y - axis

In coordinate geometry, whenever we solve a problem we see that if the resulting solution is a line, then all the lines which are parallel to y - axis are left out since their slope will be $\infty$ and thus can't be calculated. This is a big…
user2369284
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An equation for an ellipse

Definition: An ellipse is a curve on a plane surrounding two focal points such that a straight line drawn from one of the focal points to any point on the curve and then back to the other focal point has the same length for every point on the…
homogenity
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2 dimensional coordinate geometry

If $L_1$ and $L_2$ are two lines belonging to the family of lines $(3+2s)x+(4+3s)y=7+5s$ such that they are at maximum and minimum distances from the center of the circle $3x^2 +3y^2 -12x-18y-91=0$, then the equation of the lines through the point…
Geetika Meharwal
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A point can be viewed as a circle?

In analytic geometry, can a point be viewed as a circle? In analytic geometry, can the point $(0, 0)$ be view as the circle of zero radius with center $(0, 0)$?
shuxue
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Linked comprehension on straight lines

The vertex $A$ of triangle $ABC$ is $(3,-1)$. The equations of the median $BE$ and the angular bisector $CF$ are $x-4y+10=0$ and $6x+10y-59=0$ respectively. Then 1:$\;\;\;$The equation of $AB$ must be (A)…
Tejas
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Question about a pair of straight lines

Find the centroid of the triangle formed by the pair of straight lines $12x^2-20xy+7y^2=0$ and the line $2x-3y+4=0$. My doubt is: The given pair of straight lines and the third line all pass through the point $(1,2)$. So how can three concurrent…
Tejas
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Given $\vec{A_1}(1,2), \vec{A_2}(2,4), \vec{A_3}(3,b).$ find $b$ so that triangle $\triangle{A_1A_2A_3}$ will be a right-angled triangle

Given $\vec{A_1}(1,2), \vec{A_2}(2,4), \vec{A_3}(3,b).$ find $b$ so that triangle $\triangle{A_1A_2A_3}$ will be a right-angled triangle. I know that in order that $\triangle{A_1A_2A_3}$ will be right-angled , the angle $\theta$ between $A_i$ and…
Billie
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How to find out if a point lie in rectangle?

I have a rectangle in $2D$ space which is determined by $2$ points (each in opposite vertice) $p_1(x,y)$ and $p_2(x,y)$ . How can I find out numerically if a other point $p(x,y)$ is lying inside plane of the rectangle?
Emetrop
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Proving using vectors, that if a median is also a height, then the triangle is isosceles.

Proving using vectors, that if a median is also a height, then the triangle is isosceles. *Better wording would be very helpful. Thanks in advance for any help.
richard
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Some basic questions about vectors

I've got two quite basic questions about vectors. I'm sorry if it isn't right to put two questions at the same thread. I'm quite confused about the technique of solving such problems. Let $\vec v=(3,-4)$, $\vec u=(1,2)$. Find two vectors $\vec w_1,…
richard
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The locus of points that are equidistant from lines $y = x+3$ and $y = x+7$

How can I find the locus of point $P (x,y)$ that moves so that it is equidistant from lines $y = x+3$ and $y = x+7$? I take any point on the first line to be $M (x,x+3)$ and second line to be $S (x,x+7)$. When I equate PS=PM using the distance…
locuspoint
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equation of a perpendicular bisector

A diagram shown has point A( -2 , 4 ) , B ( 6, 2 ), C (-4,-4) find the equation of the line perpendicular to BC and passing through the midpoint of BC (M). Give answers in general form.
sid
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Bisector equation

I have the coordinates of the triangle vertices (A(x1,y1),B(x2,y2),C(x3,y3)) . Could I write the bisector equation taken from the top of A (AM, M(x,y))? Thanks in advance.
Liviu Solcovenco
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Family of curves $x^n+y^n=a^n$ as $n$ goes from $1$ to $\infty$ (integers) and from $1$ down to $0$

Take values of $n$ from 1 to $\infty$ in steps of $1$. Prove that in the limit it will be a square of side $a$. Take value of $n$ from $1$ to $0$ (fractions). In the limit as $n$ approaches $0$, prove that its a function which has value $a$ at $x=0$…
Babji
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