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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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Two straight lines one being a tangent to $y^2=4ax$ and the other to $x^2=4by$ are at right angles.Find the locus of their point of intersection.

Two straight lines one being a tangent to $y^2=4ax$ and the other to $x^2=4by$ are at right angles.Find the locus of their point of intersection. I tried but could not reach final answer.The tangent to $y^2=4ax$ is $y=m_1x+\frac{a}{m_1}$ and the…
Brahmagupta
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Locus of point satisfying a condition

Consider a fixed point $O$ and $n$ fixed straight lines. Through $O$ a variable line is drawn intersecting the fixed lines in $P_1,P_2,\ldots,P_n$. On this variable line, a point $P$ is taken such that $n/OP=1/OP_1+1/OP_2+\cdots+1/OP_n$. How to…
user220382
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Finding a vertex of a triangle knowing the other two and its area

I have vertix A, vertix B and the area of a triangle, and I need to find the coordinates of vertex C, knowing that it's on the bisector between the first and the third sector of the Cartesian plane. Until now I found out the length of the segment AB…
user1301428
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Of the three lines $x+\sqrt3y=0,x+y=1$ and $x-\sqrt3y=0$,two are equations of two altitudes of an equilateral triangle

Of the three lines $x+\sqrt3y=0,x+y=1$ and $x-\sqrt3y=0$,two are equations of two altitudes of an equilateral triangle.The centroid of the equilateral triangle…
Brahmagupta
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Prove that the line $PQ$ passes through a fixed point

A right isosceles triangle $AOB$ ($O$ being the origin), is such that when $AO$ and $BO$ are extended to points $P$ and $Q$ the relation $2AP.BQ=AB^2$ holds. Prove that the line $PQ$ passes through a fixed point. I tried writing some equations of…
user167045
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Intuitive Way to calculate Volume of the Solid bounded by a Plane

For a National Board Exam Review: What is the volume of the solid bounded by the plane $3x+4y+6z=12$ and the coordinate axes? Answer is $4$. I am looking for a quick and intuitive way to solve this without calculus; it's not that I have no…
james
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How to get rid of the term with $xy$?

I'm trying to put this conic on an identifiable form. $$4x^2-4xy+y^2+20x+40y=0$$ I know that the term $xy$ implies that I need to rotate the conic so that $xy$ vanishes. But I've read on some books but I couldn't figure out how to do it. It seems…
Red Banana
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Where are these choices of $A',B',C'$ for this quadratic form?

I'm studying quadratic forms: In the book I'm reading, he starts by looking at quadratic forms such as: $$\varphi (x,y)=Ax^2+2Bxy+Cy^2$$ And that given this quadratic form, one can introduce via axis rotation the new coordinates $(s,t)$ with: $$x=…
Red Banana
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Find the equations of the lines tangent to the circle $x^2+y^2=r^2$ that pass through the point $(a,0)$?

Find the equations of the lines tangent to the circle $x^2+y^2=r^2$ that pass through the point $(a,0)$. My book explains that the equation of this line is $y=m(x-a)$ and then we make the…
Red Banana
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how to prove by contradiction that any distance between a curve $x^4 - x^2 + y^4 - y^2 = 0$ and the origin is less than or equal to $\sqrt{2}$

Given a closed trajectory $x^4 - x^2 + y^4 - y^2= 0$ Prove that any distance between any point on the curve and the origin does not exceed $\sqrt2$ (ie, maximum distance from the origin to the curve is $\sqrt2$) I proved it using polar…
hmmmmm
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Slope of a line perpendicular to a line of slope $0$.

I have three points: $P = (2,5)$ $Q = (12,5)$ $R = (8,-7)$ I need to find the equation of the line thru $R$ which is perpendicular to $PQ$. How do I do this? $PQ$'s gradient is $\frac{5-5}{12-2}=0$ so the line perpendicular to that has what kind…
Deniz
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finding a soild from five planes

Given five planes: $\pi_1=2x+5y+z-2=0$ $\pi_2=x+y-z-1=0$ $\pi_3=x+4y+2z-4=0$ $\pi_4=3x-y+4z-3=0$ $\pi_5=-6x+2y-8z+k=0$. How can i find the solid shape that is formed by those planes? I tried to draw but it's too complex. I can see that for a solid…
arony
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How to Determine an Equation of a Circle using a Line and Two Points on a Circle

My question goes like this: Determine the equation of a circle tangent to the $x$-axis and passing through $(5,1)$ and $(12,8)$. I need not only the answers, but also the steps on how you did it so that I can do it on similar questions. Thanks.
Daryl
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implicit equation for elliptical torus

I just wondering what the implicit equation would be if an ellipse with major axis a and minor axis b, rotating about the Z axis with a distance of $R_0$. The $R_0$>a and $R_0$>b which means the rotation will result in a non-degenerate torus. My…
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geometry of a hyperbola and circle drawn together

how to calculate the radius of a circle which is drawn below(inwards) the hyperbola curve touching it.need a relationship between these hyperbola and circle .If a circular object is place below the hyperbola touching it how to calculate the radius…
ramya
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