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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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Locus problem need help

My try Let $(x,y)$ place on the demanded graph $\sqrt {x^{2}+y^{2}}-1=\sqrt {x^{2}+(y+3)^{2}}$ $-2\sqrt {x^{2}+y^{2}}= 6y+8$ $-\sqrt {x^{2}+y^{2}}=3y+4$ $x^{2}+y^{2}= 16+9y^2+24y$ What should I do now?
user373141
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Distance from point to cone

"Consider the cone $$ C := \{ (x,y,z) \in \mathbf{R}^{3} : x^{2}+y^{2} = z^{2}\}$$ and the point $ P:=(1,2,3) $. Find the least diatance between $P$ and any point $Q\in C$ and the coordinates of this point." Here's what I did. Project everything…
wet
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Proof that $\dfrac{\|\vec{CA}\|}{\|\vec{AX}\|} = \dfrac{\|\vec{CB}\|}{\|\vec{BX}\|}$

Proof that$$\dfrac{\|\vec{CA}\|}{\|\vec{AX}\|} = \dfrac{\|\vec{CB}\|}{\|\vec{BX}\|}$$ having $$\dfrac{\vec{CA}}{\|\vec{CA}\|} + \dfrac{\vec{CB}}{\|\vec{CB}\|} = \alpha\vec{CX}$$ I have tried to inject the second formula in the first, but I wasn't…
Carlos Oliveira
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Find the equation of straight lines through the point $(\dfrac {1}{\sqrt {3}}, 1)$ whose perpendicular distance from the origin is unity.

Find the equation of straight lines through the point $\left(\dfrac {1}{\sqrt {3}}, 1\right)$ whose perpendicular distance from the origin is unity. My Attempt: Let the equation of line be $ax+by+c=0$. Its distance from origin is $1$ unit.…
pi-π
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Find the area of a triangle using analytic geometry

Given are the points $P (1,0)$ and $Q (3,2)$. The points $P$ and $Q$ have the same distance to a certain line $l$, which intersects the positive x-axis in the point $A$ and the positive y-axis in the point $B$. The area of the triangle $ABO$ is…
JohnPhteven
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closest distance between a line and a lattice point in the plane

What is the closest distance between the line $y= 4/7 x + 1/5$ and a lattice point in the plane. Here is my work: I re-wrote the equation of the given line as $-20x +35y -7=0$ Then Let $I(a, b)$ be the closest lattice point to the given line. $d=…
Big Boy
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Find out for which values of $\lambda$ the points of the line are inside the circle

We have a line (in parameter): $ x = 2\lambda $ $ y = 1-\lambda$ Find out for which values of $\lambda$ the points of the line are inside the circle of $x^2+4x+y^2-6y+5=0$ What I did: I rewrote the circle to the form $(x+2)^2 + (y-3)^2 = 8$. Where…
JohnPhteven
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Maximum number of common normal of $y^2=4ax$ and $x^2 = 4by$

Maximum number of common normal of $y^2=4ax$ and $x^2 = 4by$ Attempt: Equation of normal to the curve $y^2=4ax$ in slope form $y=mx-2am-am^3.$ Now above equation is also normal to $x^2=4by$ I could not understand how to solve further, thanks.
DXT
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Reflection over a line $y= ax + b$ - How to find the according point?

Let's take a point $E = (x_A, y_A)$ and line $p: Ax+By+C=0$ Let's find the reflection point of point $E$ over the line $p$ I find the equation of a line perpendicular to $p$ and passing through point E I find the distance between $E$ and $p$:…
ILoveChess
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Solution of a logarithmic and an exponential curve

What is the least positive integral value of $k$ such that the equation $$\ln x + k = e^{x-k}$$ has a solution? I tried plotting the two curves and concluded that the limiting value of k will be the case when $y = \ln x +k$ and $y = e^{x-k}$…
Sourav Suman
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Find the point on OY

The task is: Given points $M(-3,2)$ and $N(2,5)$. Find a point $P$ on $OY$ that $|MP-NP|$* is maximum.(You are not allowed to use anything(vectors) except the general equation of straight line and the definition of the distance between two points) I…
ModOverRing
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Finding Locus - Parabola

In the point A on the parabola: $y^{2}=2px$ a tangent line to the parabola is being drawn such that it meets the y-axis at the point B. Find the locus of the meeting points C of the line from B which is parallel to the x-axis with the line from A…
user3275222
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Analytic Geometry: line parallel to a plane

I'm trying to figure out how to solve this problem of Analytic geometry of the space: I have a line described by the system: $$\begin{cases} 2x-y+z-1 &=0\\ 5x+3y-8& =0 \end{cases}$$ and I have to verify wether this line is parallel or not to the…
FET
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Maximum distance of $P$ from side $BC$

Let $P$ be a point which doesnot lie outside the triangle $ABC$,$A(3,2),B(0,0),C(0,4)$ and satisfies $$d(P(A))\geq \text{ }max [d(P,B),d(P,C)] $$then the question is to find out the maximum distance of $P$ from side $BC$ where $d(P,A)$ gives the…
Navin
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A variable line intersecting $n$ concurrent straight lines

Problem Statement:- A variable line cuts $n$ given concurrent straight lines at $A_1,A_2,A_3,\ldots A_n$, such that $\displaystyle\sum_{i=1}^{n}{\frac{1}{{OA}_{i}}}$ is a constant. Show that it always passes through a fixed point, $O$ being the…
user350331
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