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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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Through the point $A(4,5)$ a line is drawn.

Through the point $A(4,5)$ a line is drawn inclined at $45°$ with the $+ve$ X - axis. It meets $x+y=6$ at the point $B$. Find the equation of $AB$. My solution.. Equation of $AB$ $$(y-y_1)=m(x-x_1)$$ $$(y-5)=1(x-4)$$ $$x-y+1=0$$. But the answer in…
Iaamuser user
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Normal to the plane under the condition describes the cone

The plane $lx+my+nz=0$ moves in such a way that its intersection with the planes $ax+by+cz+d=0$ and $a'x + b'y + c'z+d'=0$ are perpendicular. Show that the normal to the plane through the origin describes in general, a cone of the second degree and…
Soham
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A question about an equation of a plane

Let $A=(1,3,1)$; $B=(1,1,1)$; $C=(2,0,1)$; $D=(1,-2,3)$. Determine the equation of a plane that passes through $D$ and is parallel with $(ABC)$. I know the fact that $\mbox{dir}(\text{plane})=\mbox{dir}(ABC)$ if plane is parallel to $(ABC)$, but I…
Adrian Costea
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Graphing a regular pentagon

I just realized that I didn't know how to graph a regular pentagon with integer coordinates... What are some possible coordinates for a regular pentagon with the uppermost point at coordinate (0,0)?
suomynonA
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Draw a plane through a line parallel to the $x$-axis

Can someone help me with this problem? I bet it will be pretty easy for the most of you: Through the line $p$ draw a plane that is parallel to the $x$-axis, where p is defined by: $x=5-2t, y=9t-1, z=-7t-2$. Since it's parallel to the $x$-axis,…
CHaOS
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In $\triangle ABC, PA+PB+PC=AB+AC, PE=x, f(x)=x+BE+CE, $then $f(x)$ is mono increasing function

In $\triangle ABC, PA+PB+PC=AB+AC, PA$ extend to cross $BC$ at $D, E$ is on $PD$, let $PE=x,f(x)=x+EB+EC$, then $f(x)$ is mono increasing function. I can write $BE=\sqrt{AB^2+(AP+x)^2-2AB*(AP+x) \cos \alpha} $ $CE=\sqrt{AC^2+(AP+x)^2-2AC*(AP+x)…
chenbai
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I want to find 3 planes that each contain one and only one line from a set

The three lines intersect in the point $(1, 1, 1)$: $(1 - t, 1 + 2t, 1 + t)$, $(u, 2u - 1, 3u - 2)$, and $(v - 1, 2v - 3, 3 - v)$. How can I find three planes which also intersect in the point $(1, 1, 1)$ such that each plane contains one and only…
John
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Find the equation of the line which is

Find the equation of the line perpendicular to the line joining the points $A(3,6)$ and $B(-6,9)$, which divides the line $AB$ in the ratio of $2:1$. My attempt: Equation of $AB$ is $$y-y_1=\frac{y_2-y_1}{x_2-x_1} (x-x_1)$$ $$y-6=\frac{9-6}{-6-3}…
Iaamuser user
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Prove that one of the lines represented by $ax^2+2hxy+by^2=0$ will bisect the angle between the coordinate axes if $(a+b)^2=4h^2$.

Prove that one of the lines represented by $ax^2+2hxy+by^2=0$ will bisect the angle between the coordinate axes if $(a+b)^2=4h^2$. Solution I calculated the two lines represented by $ax^2+2hxy+by^2=0$ as…
Iaamuser user
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Equation of a Pair of Straight Lines .2nd degree

Show that if one of the lines given by $a_1x^2+2h_1xy+b_1y^2=0$ coincides with one of the lines of $a_2x^2+2h_2xy+b_2y^2=0$ then $(a_1b_2 - a_2b_1)^2=4(a_2h_1 - a_1h_2)(b_1h_2-b_2h_1)$ Actually, I did not get any idea to start its solution. so…
Iaamuser user
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Finding the equation of diagonal

If $ax^2+2hxy+by^2=0$ be the two sides of a parallelogram and $px+qy=1$ is one diagonal then prove that the other diagonal is $y(bp-hq)=x(aq-hp)$. My solution is here; $ax^2+2hxy+by^2=0$ Multiplying by a and adding h^2y^2,…
Iaamuser user
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Equation of a plane given one point and two planes

I've done a question similar to this, however this one has no complete equations i can solve for. Determine the equation of the plane that passes through $(1,3,8)$ and is perpendicular to the line of intersection of the planes $3x-2z+1=0$ and…
Saad Siddiqui
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Homogeneous Equation of second degree in x and y.

Find the single equation of the two lines through the origin and perpendicular to the each lines represented by $ax^2+2hxy+by^2=0$ I tried the factorization of the given equation but it was fail..
Ger Wyn
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Finding distance of points of intersection of curve with another form its center.

Let $C$ be a curve which is locus of point of intersection of the lines $x=2+m$and $my=4-m$. A circle $S:(x-2)^2+(y+1)^2=25 $ intersects the curve $C$ at four points $P,Q,R,S$. If $O$ is the centre of the curve $C$ then find $OP^2+OQ^2+OR^2+OS^2$. I…
mathemather
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Finding the equation of locus

The co ordinates of any position of a moving point P are given by $$\left[\frac{(7t-2)}{(3t+2)} , \frac{(4t+5)}{(t-1)}\right]$$ where $t$ is a variable parameter.Find the equation of locus of $P$. I could not understand the question especially…
Ger Wyn
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