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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.

6689 questions
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About a tangent line

We know that if $y=f(x)$ and $f'(x)$ exists at $x=x_1$ then the equation of the line tangent at the point $(x_1,f(x_1)$ is given by $$y-f(x_1)=f'(x_1)(x-x_1).$$ My question is this. What if $x=g(y)$ and $g'(y)$ exists at $y=y_1$. What would be the…
Juniven Acapulco
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Functions $y = x^2 + x - 1$ and $y = x^3 + 2x^2 + (a + b\sqrt{3})x - 3$ have three common points $A, B, C$ such that the circumradius is $R = 3$.

Consider two functions $y = x^2 + x - 1$ and $y = x^3 + 2x^2 + (a + b\sqrt{3})x - 3$ with $a$ and $b$ being two rational numbers such that the graphs of the aforementioned functions share three common points $A, B, C$ such that the radius of the…
Lê Thành Đạt
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The area between coordinate axes and tangent to $f(x)$ is $2x f(x)$. Is this correct?

I was checking the answers of the coordinate axes has minimum area question. In short, the mentioned question asks the minimum area formed by a tangent to $y=4-x^2$ and coordinate axes. I noticed one of the answers to the question mentioned an…
Kartal Tabak
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Minimization problem of $e^x$ and straight line.

Given the curve $C$: $y=e^{x}$ and a point $P$: $(x_0,y_0)$ with $y_0 > e^{x_{0}}$, a straight line $L$ of slope $k$ goes through the point $P$ and intersects $C$ at two points $A$ and $B$, try to find out: $(1).$ $k$ that minimizes the distance of…
peterwang
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Proof of the Inscribed Angle Theorem using vectors

I am trying to prove the inscribed angle theorem using vectors. I fixed the dots $A=(\cos\theta,\sin\theta)$, and $B=(\cos\varphi,\sin\varphi)$, and I took another point $C=(\cos\psi,\sin\psi)$ in the biggest arc $AB$. My idea was to calculate…
Lorenzo Andreaus
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Locus of points $(a,b)$ such that circle $(x-a)^2+(y-b)^2=b^2$ is tangent to parabola $y=x^2$

Let $P \colon\; y = x^2$ be a parabola and $C_{a,b} \colon\; (x-a)^2 + (y-b)^2 = b^2$ be a circle. Suppose that $C_{a, b}$ contacts the parabola $P$ with one point, i.e., $C_{a, b}$ tangential to $P$. Is it easy to describe the locus of the center…
Rinmyaku
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Is there a rational parametrization of Quadric surfaces?

Does there exists a rational parametrization of quadratic surfaces? In particular, I want to parametrize hyperboloid of one sheet $\frac{x^2}{b}+\frac{y^2}{4b}-\frac{z^2}{4b}=1$ where $b$ is rational. (https://en.wikipedia.org/wiki/Hyperboloid). In…
ersh
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The difference between the radii of the largest and smallest circles having centres on the circumference of $x^2+2x+y^2+4y=4$

Owing to restriction of 150 characters in the title section I include the latter part of the problem here below in bold and italics Also given that both the circles(largest and smallest) pass through a point $(a,b)$ lying outside the given circle,…
Saradamani
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Equilateral triangle inscribed in an ellipse

An equilateral triangle of side length $ L\approx 6.14$ and one side inclination $ \approx 49.52^{\circ}$ is inscribed in an ellipse of semi-axes $(a,b) = (5,3)$. Drawn in Geogebra by Java mousing .. trial/error. Are $ (L,\alpha) = f(a,b) $ in…
Narasimham
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Distance minimum distance between point and sphere

How can I find minimum distance between point and sphere ? sphere properties : position of center a,b,c redius of the sphere R point properties position x,y,z
jques
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3 normals to a parabola

What is the least value of $z$ such that 3 normals from $P(z,0)$ can be drawn to $y^2=4ax$? I thought that any point inside the parabola should satisfy this condition but I was proven wrong.
palash jain
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Help in understanding a simple proof.

Let $Ax + By + C = 0$ be a general equation of a line and $x\cos \alpha + y\sin \alpha - p = 0$ be the normal form of the equation. Then, $${-p\over C } = { \cos \alpha\over A} = { \sin\alpha\over B}\tag{1}$$ $${p\over C } = { \cos \alpha\over -A}…
user312097
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How prove this $BC$ always passes through a fixed point with $\frac{x^2}{4}+y^2=1$

if the point $A(0,1)$ on the ellipse $\Gamma:$ $\dfrac{x^2}{4}+y^2=1$ and the circle $\tau:$ $(x+1)^2+y^2=r^2(0
math110
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What is the coordinate of a point $P$ on the line $2x-y+5=0$ such that $|PA-PB|$ is maximum where $A=(4,-2)$ and $B=(2,-4)$

What is the coordinate of a point $P$ on the line $2x-y+5=0$ such that $|PA-PB|$ is maximum where $A=(4,-2)$ and $B=(2,-4)$. Let the coordinates of the point $P$ be $(x_1,y_1).$ $P$ lies on the line $2x-y+5=0$,so…
user1442
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Find equation for circle related to a triangle

write the equation of the circle of the triangle with vertices $$A = (5 ,\ -4 ),\ B = (6 ,\ -1 ),\ C = ( 2,\ 3)$$ examine the relative position of this district and its image in axial symmetry about the line $$3x + 4y + 26 = 0$$ I think about this:…
Jon.Don
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