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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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How to calculate the asymptotes to $y^2+2y-4x^2=0$

Calculate the center and asymptotes to the hyperbola $y^2+2y-4x^2=0$, aswell as the intersections of the actual coordinate axis. The hyperbola takes the form $\left ( \frac{x}{\frac{1}{\sqrt{4}}}\right )^{2}-\left ( {y+1}\right )^{2}=-1$ and I…
Andreas
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Locus of mid point of $AB$

If the family of lines $tx+3y-6=0.$ where $t$ is variable intersect the lines $x-2y+3=0$ and $x-y+1=0$ at point $A$ and $B.$ Then locus of mid point of $AB$ is what i try Intersection of line $tx+3y-6=0$ and $x-2y+3=0$ is $\displaystyle…
jacky
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Max. distance of Normal to ellipse from origin

How Can I calculate Maximum Distance of Center of the ellipse $\displaystyle \frac{x^2}{a^2}+\frac{y^2}{b^2} = 1$ from the Normal. My Try :: Let $P(a\cos \theta,b\sin \theta)$ be any point on the ellipse. Then equation of Normal at that point…
juantheron
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The $x y=c^2$ problem

If the points $P$, $Q$, $R$ satisfy the relation $x y=c^2$, then prove that the orthocentre formed by these $3$ points also satisfies the above relation.
ssk
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Figuring out the Orthogonal Coordinates from a a given point and vector

I am given this Question of Gandalf the Grey started in the Forest of Mirkwood at a point with coordinates (-2, -3) and eventually arrived in the Iron Hills at the point with coordinates (0, 0). If he began walking in the direction of the vector…
user591094
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How to find asymptotes of this weird hyperbola

How does one find equations for the asymptotes of hyperbolae of the form: $$k \mu^2-2cq\mu-\sigma^2d+q^2z=0$$ where $\mu$ is the dependent variable, $\sigma$ is the independent variable, and the rest are parameters.
ben
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Show that the two straight lines $x^2(\tan^2 (\theta)+\cos^2 (\theta))-2xy\tan (\theta)+y^2.\sin^2 (\theta)=0$

Show that the two straight lines $x^2(\tan^2 (\theta)+\cos^2 (\theta))-2xy\tan (\theta)+y^2.\sin^2 (\theta)=0$ make with x axis such that the difference of their tangents is $2$. My Attempt: $$x^2(\tan^2 (\theta) +\cos^2 (\theta))-2xy\tan (\theta) +…
pi-π
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How many step I need to (99,99,19)?

The start point is (0,0,0). The "step vector" is (2,3,4). I can variate this, so it can be (3,4,2) and so on, and it can be negative too, like (4,2-,3). What is the minimum step to reach the finish (99,99,19) point, if I can? And with other step…
justgiveup
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Find $k$ such that the intersection of $x+ky=1$ and $y^2 - x^2 - z^2 = 1$ is an ellipse or a hyperbola

Find the values of $k$ such that the intersection of the plane $x+ky=1$ with the two-sheeted elliptic hyperboloid $y^2 - x^2 - z^2 = 1$ is (a) an ellipse and (b) a hyperbola. My attempt is the following: If I choose a particular value for $x$ such…
favq
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Find a plane whose intersection line with a hyperboloid is a circle

Find a plane $\pi$ which involves x-axis and its intersection line with $$\frac{x^2}{4}+y^2-z^2=1$$ is a circle. Because the plane want to be find involves x-axis,so set as $By+Cz=0$,then I must to determine its value such that…
Laura
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Product of equation of two lines

Why does the product of equations of straight lines represents 2 straight lines? Why does a 2nd degree equation is so linear?
Kush
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Hyperbolas on an Imaginary Graph

My first question is what this type of graph (of $x-y-i$) is called since I was unable to find any information about any such graph. Now for the real question, I used the equation $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ and drew the graph on the…
Rohan Verma
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The angle between the asymptotes of a hyperbola is $\frac{\pi}{3}$. Determine the eccentricity of the hyperbola

Would appreciate if someone can help me with this one. The angle between the asymptotes of a hyperbola is $\dfrac{\pi}{3}$. Determine the eccentricity of the hyperbola. Since I'm trying to self teach myself here, the only thing I could find was…
Edward B
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Need proof for a property in Analytic Geometry

When I was taught Vector Calculus, we assumed the following properties of the dot product (without proof): $$\vec{\mathbf{a}} \cdot \vec{\mathbf{b}} = \vec{\mathbf{b}} \cdot \vec{\mathbf{a}}$$ $$\vec{\mathbf{a}} \cdot (\vec{\mathbf{b}} +…
Truth-seek
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Generators of hyperbola

A variable generator meets two generators of the system through extremities $B$ & $B'$ of the minor axis of the principal elliptic section of the hyperboloid $$\frac{x^2} {a^2} +\frac{y^2}{b^2} -z^2c^2=1$$ in $P$ & $P'$. Prove that $BP$.…
aarbee
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