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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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Planar equation with missing variable

A planar equation with one missing variable, i.e. $ax+bz-d=0$ has shown in my Math exam. Is this a valid planar equation? It seems to me as $y = 0$, this would produce only a straight line.
Pedro Sobota
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Equation of a certain region inside a regular polygon of $n$ .

Consider a regular polygon $A_1A_2....A_n$ of n sides with unit side length.Without loss of generality we can take $A_1$ coincident with the origin O and $A_2$ on positive x axis s.t. $A_1A_2=1$.If $D$ in the region in the polygon with the…
AgnostMystic
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An affine transformation of a plane onto another plane

How to write down the affine transformation that transforms the plane $Ax+By+Cz+D=0$ onto the plane $z = 0$? I cannot understand how to begin. I can express $z$ (if $C\neq 0$): $$ z=-\frac{D+Ax+By}{C} $$ And I need to get the plane $z = 0$. How to…
KateCh
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The equation of one diagonal of a square is $2x+3y=5$..

The equation of one diagonal of a square is $2x+3y=5$ and the coordinates of one vertex is $(1,-3)$. Find the equations of two sides of the square which pass through the given vertex. My Attempt: Given, $$2x+3y=5$$ Slope of given line is $-2/3$. And…
pi-π
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Vector VS Plane intersection

Could You help me with task: From point $M(3,5)$ that belongs to plane: $A(0,0), B(0,10), C(20,10), D(20,0)$, comes out vector $V$ at an angle a(with $OX$). Need to find point $X(x,y)$ at which he will be leaving the plane. Thanks.
mathJunior
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What is the radius of the circle $B$?

If one of the diameters of the circle $A$ with equation $x^2+y^2-2x-6y+6=0$ is a chord to the circle $B$ with centre $(2, 1)$ then the radius of the circle $B$ is $(1)\ \sqrt 3$ $(2)\ \sqrt 2$ $(3)\ 3$ $(4)\ 2$ If I draw a perpendicular from $(2,1)$…
Arnab Chattopadhyay.
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Find a point $P$ on a curve given the gradient of the normal at $P$.

The point $P$ lies on curve $y=(x-5)^2$. It is given that the gradient of the normal at $P$ is $-\frac 14$ Find the coordinates of $P$.
bryan
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Find x-intercept of y=15 on huge ordered pair

I've got the ordered pairs $(1491947996, 15.7)$ and $(1491948898, 12.9)$. The X values in each set indicate the seconds from Unix Epoch and the Y values in each set indicate the temperature in Celsius. I'm trying to determine the X-coordinate where…
Jon Mitten
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Equation of common tangent

Equation of common tangent of circle $x^2 + y^2 -8x =0$ and hyperbola with $x^2/9 - y^2/4 =1.$ I tried this but was left with lengthy solutions can u please explain with easy solutions.
palash jain
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Direct/transverse tangents of two circles

If equations of two circles in a plane are given as $$ f(x,y)=0,\, g(x,y)=0,\, $$ Find equations of direct tangents in terms of $f,g$ and Find equations of transverse tangents in terms of $f,g$ We could calculate using coincident roots/point…
Narasimham
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If $Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$ , why $cot(2\theta) =\dfrac{{A}-{B}}{C}$ in conic sections?

I would like to know why $$ cot(2\theta) =\dfrac{{A}-{B}}{C} $$ given $$ Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 $$ Thank You!
user2860452
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Finding line of intersection of two planes

When an equation of line is given as intersection of two planes like $$ax + by + cz + d = 0 = px + qy + rz + s,$$ why do we put $z=0$?
novice
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The line $y=mx+c$ is a tangent to $x^2+y^2=a^2$ if:

The line $y=mx+c$ is a tangent to $x^2+y^2=a^2$ if: $1$. $c=a\sqrt {1+m^2}$ $2$. $c=\pm a\sqrt {1+m^2}$ $3$. $c^2=\pm a\sqrt {1+m^2}$ $4$. $\textrm {None}$ My Attempt: The tangent to circle $x^2+y^2=a^2$ at point $(x_1,y_1)$ is given by…
pi-π
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Solving an equation for an ellipse

It's some years since I did math and I can't, for the life of me remember how to solve equations. I am trying to find the centre of an ellipse from the radii and $2$ points on the ellipse using : $$\frac{(x_A-C_x)^2}{a^2} + \frac{(y_A-C_y)^2}{b^2} =…
Dave Coventry
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The max area of a tangential parallelogram of an ellipse

Given an ellipse $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$, $a > b>0$ find out the minimum area of its tangential parallelogram, and specify when this max area is achieved.
Pin Yin
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