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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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Finding the equation of a coaxial circle with its diameter falls on the radical line

Here is the problem:- $L: x – y + 3 = 0$ is the radical line for $S$, the system of coaxial circles. $C: x^2 + y^2 – 2x – 4y – 11 = 0$ is a member of $S$ with $AB$ as the common chord. (a) Find the equation of $T$, the line of centers of $S$. (b)…
Mick
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Finding vertices of rhombus formed by lines $y=2x+4$, $y=-\frac{1}{3}x+4$ and $(12,0)$ is a vertex. Can't find last vertex.

The equations of two adjacent sides of a rhombus are $y=2x+4$, $y=-\frac{1}{3}x+4$. If $(12,0)$ is one vertex and all vertices have positive coordinates, find the coordinates of the other three vertices. (Need help finding last vertex) I know that…
shredalert
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I was taught that the tangent of the angle between two lines having slopes $m_1$ and $m_2$ has modulus of $\frac{m_1-m_2}{1+m_1m_2}$.

I was taught that the tangent of the angle between two lines having slopes $m_1$ and $m_2$ is given by the formula modulus of $\frac{m_1-m_2}{1+m_1m_2}$. We can then use $\tan^{-1}$ to find the angle. However, some angles have negative tangent…
N.S.JOHN
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Showing boundedness of a set defined by equality

I am trying to show that the set given by: $$S = \{\mathbf x \in \Bbb R^2 \mid x^2 + 3xy + 3y^2 = 3\}$$ is bounded. I am able to show that this is true whenever $(x,y) \in \Bbb R^2$ is such that: $x>0$ and $y>0$; or $x<0$ and $y<0$. However, I…
bluemackillop
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Equation of a Straight lines.

A variable straight line drawn through the point of intersection of the straight lines $\frac xa + \frac yb=1$ and $\frac xb + \frac ya=1$ meets the co ordinate axes at $A$ and $B$. Prove that the locus of the mid point of $AB$ is…
Ger Wyn
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Equation of Locus of a point

For the variable triangle $ABC$ with the fixed vertex at $C(1,2)$ and $A,B$ having co ordinates $(\cos t, \sin t)$, $(\sin t, -\cos t)$ respectively, find the locus of its centroid. Plz help me, I could not even get how to start.
Ger Wyn
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Equation of a Pair of Straight Lines.

If $ax^2+2hxy+by^2$ be the two sides of the parallelogram and $px+qy=1$ is one diagonal then prove that the other diagonal is $y(bp-hq)=x(aq-hp)$. By reading the question I just understood that the given diagonal is not through origin. Second…
Ger Wyn
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finding the equation of tangent lines to curves

Find the equation of the tangent to the curve $y = x^2 -6x +5$ at each point where the curve cuts the axis. Find also the coordinates of the point where these tangent line meet. I found the gradient function to be $2x-6$ and I know the curve cuts…
Tesha Caesar
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Find the equations of Lines given Points and Angle

I have the following scenario. The coordinates of points B and D are $(10,0)$ and $(10,-10)$ respectively. I want to construct angle $\angle BFD = 45^\circ $. How can i find the coordinates of point F in order to construct this angle? In other…
KeyC0de
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finding the coordinate given a distance with its coordinates

A point $P(x,y)$ has a distance $5\sqrt{2}$ units from $Q(4, -7)$ and a distance $\sqrt{106}$ units from $R(-6,5)$. Knowing that, find $P$. the image is exactly the set of problem that our professor given to us I assume that the 3 problems are…
justine
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Number of places two intersecting lines can intersect a hyperbola

If each of two intersecting lines intersects a hyperbola and neither line is tangent to the hyperbola,then what are the possible numberof places where the lines can intersec the hyperbola ? I've played a bit with GeoGebra and it seems to me…
Mr. Y
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How to rotate an hyperbola by $45^\circ$ so that I have an equation of the form $xy=c$

I am trying to show that If I rotate an hyperbola of the form $\cfrac{x^2}{a^2}-\cfrac{y^2}{b^2}=1$ by $45^\circ$ I get an equation of the form $x'y'=c$. Using the following rotation coordinates: $x=x' \cos 45^\circ -y' \sin 45^\circ$ $y=x' \sin…
Mr. Y
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How to get rotation coordinates of point $(x',y')$ in terms of $(x,y)$

Given that we have $x=x' \cos \theta - y' \sin \theta $ and $y=x' \sin \theta +y' \cos \theta $ ,how can I express $x',y'$ in terms of $x,y$ and $a$ ? I've browsed through the site to seek for some help but I've found that most of the questions…
Mr. Y
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Sphere centered in line $(x,y,z) = (-2,0,1) + \lambda (0,0,1)$ tangent to planes $ x-10z = 0 $ and $ x+2z = 0 $ whose radius squared is $r^2 > 20$

how may I find the sphere centered in line $$(x,y,z) = (-2,0,1) + \lambda (0,0,1)$$ tangent to planes $$ x-10z = 0 $$ and $$ x+2z = 0 $$ whose radius squared is $$r^2 > 20$$ Thank you.
bru1987
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Ellipse by moving center of a parametric circle equation?

Given that a parametric eq for a circle is given by : $$x= r \cos \theta \\ y= r \sin \theta $$ Is it possible to move the center of circle by a (periodic) function $f(r,\theta)$: $$\begin{align} x &= r \cos \theta + f(r,\theta)\\ y &= r \sin…
jimjim
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