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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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Determining the Diameter of a Circle

The radius of a circle can in one way be determined if we know its Diameter $D$. It can be determined by the equation: $r = \frac D2$ ..... (I) We can also determine the radius of a circle by Euclidean distance formula which is as follows if the…
Samama Fahim
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Distance between the two points $P$ and $Q$

Let $d(x,y)$ denote the distance between two points, $x$ and $y$, on the plane. 1) $P(2,9),\quad Q(-1,13)\Rightarrow d(P,Q) = 5.$ 2) $P(1,-2),\quad Q(2,10)\Rightarrow d(P,Q) = \sqrt{145}.$ 3) $P(0,0),\quad Q(-2,-3)\Rightarrow d(P,Q) = \sqrt{13}.$ 4)…
Samama Fahim
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Properties of a line lying wholly on a surface

Why is the coefficient of $r^2$ and $r$ equal to zero if a line lies wholly on a surface? (see pics)
peaceHoper
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Show that when $p$ varies, the midpoint of $PR$ lies on the curve $y^2=2x-4$.

The line passing through the point $P$ ($p^2,2p$) on the curve $y^2=4x$ and the point $Q$(2,0) intersects the curve once again at point $R$, find the coordinates of point $R$ in terms of $p$. I am able to solve this part of the question and found…
gc3941d
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The equation of $AB$ is $y= 2x-6$ The equation of $AC$ is $y= -x+12$. Calculate the size of the angle $A$

I have tried various methods as to answer this question using $\sin$ and $\tan$ but I cannot seem to attach my working out please send me the answers no working out needed as I just want to see where I went wrong thank you
zee
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How to compute minimum distance of parallel lines to contain a unit square in a grid at any offset?

Given two parallel lines placed on graph paper whose angle relative to vertical is θ, where θ is less than or equal to 45°, what is the minimum distance between the two lines to guarantee that there is at least one grid of the graph paper that…
WilliamKF
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If a circle intersects the hyperbola $y=1/x$ at four distinct points $(x_i,y_i), i=1,2,3,4,$ then prove that $x_1x_2=y_3y_4$.

If a circle intersects the hyperbola $y=1/x$ at four distinct points $(x_i,y_i), i=1,2,3,4,$ then prove that $x_1x_2=y_3y_4$. I have really no idea on how to approach this question. One clumsy way might be to consider an arbitrary circle…
Sanket Biswas
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Show that a necessary and sufficient condition is either $ac<0$ or $bc<0$.

Question: Show that a necessary and sufficient condition for the line $ax+by+c=0,$ where $a,b,c$ are nonzero real numbers, to pass through the first quadrant is either $ac<0$ or $bc<0$. Solution: We have $ax+by+c=0,$ where $a,b,c$ are nonzero real…
Sanket Biswas
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The calculation of an Dual norm

I computing dual norm $ \|v\|_\ast = \sup_{\|x\|\leq1} \langle v, x\rangle $. For example, I process like this: $x_1 = (2, 1); x_2 = (5, 10); x_3 = (8, 10)$. As norm of $x$ must be $\|x\| \leq 1$, I normalized vectors and I get this : $ x_1 =…
K.J Fogang Fokoa
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Finding value of radius in inscribed circle relationships

I got this question on a grade 11 multiple choice test and had no idea how to solve it. Q: In the diagram, the circle with centre X is tangent to the largest circle and passes through the centre of the largest circle. The circles with centres Y and…
Sinestro 38
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A problem related to normal to a parabola.

Let $P$ be the point on the parabola $y^2 = 3x$ such that $OP$ makes an angle of $\pi/6$ with the $x$-axis, where $O$ is the origin. A normal is drawn to the parabola at $P$, intersecting the axis of the parabola at $Q$. If $S$ is the focus of the…
Normal
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Finding points on a line a given distance from a given point

The line $l$ has gradient $-2$ and passes through $A=(3,5)$. $B$ is a point on $l$ such that $\overline{AB}=6\sqrt5$. Find coordinates of each possible points of $B$. There are two $x$-coordinates and two $y$-coordinates and so far I have got to…
MathsHelp
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If the circumcentre of a triangle lies at (0,0) and centroid is the middle point of $(a^2+1,a^2+1)$ and $(2a,-2a)$.

Find the equation of the line on which the orthocenter lies The centroid G is $$G=(\frac{a^2+2a+1}{2},\frac{a^2-2a+1}{2})$$ Since it divides O(circumcentre) and H(orthocentre) in the ratio 2:1 Let the orthocentre be…
Aditya
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$\tan \frac{\alpha}{2} \tan \frac{\beta}{2}=\frac{1-e}{1+e}$

If P is a point of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ Whose foci are S and S’.Let angle PSS’=$\alpha$ and $PS’S=\beta$ then prove that $\tan \frac{\alpha}{2} \tan \frac{\beta}{2}=\frac{1-e}{1+e}$ I know that S(ae,0) and S’(-ae,0) and…
Vinod Kumar Punia
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The process of proving statements in analytic(coordinate) geometry.(to prove smthng for all quadrants.)

I have some misunderstanding with the process of proving statements in coordinate geometry. Here what I mean: In textbooks when it comes to proving something, the general process of proving (most of the time) happens in only one quadrant. Authors…
Tim Solnze
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