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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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If $PQ$ subtends right angle at the centre of ellipse then find $\frac{1}{OP^2}+\frac{1}{OQ^2}. $

$PQ$ is a variable chord of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ . If $PQ$ subtends right angle at the centre of ellipse then find $\frac{1}{OP^2}+\frac{1}{OQ^2}. $ Two points can be taken…
mathemather
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Finding an angle inside a regular hexagon given a line that passes through two vertices

I am asked to find the angle $\alpha$ on this particular setup: The equation of the line is and point $A = (2,1,4)$. Is that really possible to find out? Maybe there is some data missing? Thank you.
bru1987
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finding equation of the normal

Find the equation of the normal to the curve $y= 4/x$ at point where $y=1/2$. Find the coordinates of the point where this normal cuts the x axis. I know that the curve cuts the x axis when $y=0$ and I tried doing that for starters.
Tesha Caesar
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Distance between points - equation of a line

I have worked on this particular example: The distance between the point $M_1(3,2)$ and $A$ is $2\sqrt5$ and the distance between the point $M_2(-2, 2)$ and $B$ is $\sqrt5$. Come up with a equation for the line going through $A$ and $B$. I have…
n4869
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Let points $A,B,C$ are represented by $(a\cos\theta_1,a\sin\theta_1),(a\cos\theta_2,a\sin\theta_2),(a\cos\theta_3,a\sin\theta_3)$ respectively

Let points $A,B,C$ are represented by $(a\cos\theta_1,a\sin\theta_1),(a\cos\theta_2,a\sin\theta_2),(a\cos\theta_3,a\sin\theta_3)$ respectively and $\cos(\theta_1-\theta_2)+\cos(\theta_2-\theta_3)+\cos(\theta_3-\theta_1)=\frac{-3}{2}$.Then prove that…
user1442
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$A$ is a point on either of the two lines $y+\sqrt3|x|=2$ at a distance of $\frac{4}{\sqrt3}$ units from their point of intersection.

$A$ is a point on either of the two lines $y+\sqrt3|x|=2$ at a distance of $\frac{4}{\sqrt3}$ units from their point of intersection.Find the coordinates of the foot of perpendicular from $A$ on the bisector of the angle between them? The two lines…
user1442
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Two point form of solving straight line problems in coordinate geometry

If we find the slope of a line via two point form we find that when the for different points in the same straight line we have different equations. Why is it so? Say, if the points were $3,5$ and $6,10$ then the equation was $2x-y=1$ and when the…
Arj
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Let $M$ be the locus of the feet of the perpendiculars drawn from the points on $L$ to the plane $P_1$

In $R^3$,let $L$ be a straight line passing through the origin.Suppose that all the points on L are at a constant distance from the two planes $P_1:x + 2y-z + 1 = 0$ and $P_2 : 2x-y + z-1=0.$ Let $M$ be the locus of the feet of the perpendiculars…
diya
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Let $m_1$ and $m_2$ are the slopes of the tangents drawn to circle $x^2+y^2-4x-8y-5=0$ from the point $P(-1,-2)$,and $|m_1+m_2|=\frac{p}{q}$

Let $m_1$ and $m_2$ are the slopes of the tangents drawn to circle $x^2+y^2-4x-8y-5=0$ from the point $P(-1,-2)$,and $|m_1+m_2|=\frac{p}{q}$,where $p$and $q$ are relatively prime natural numbers,then find $p+q.$ Slope of…
Vinod Kumar Punia
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Prove that normals at the points where line intersects a parabola meet at point on normal at a given point on the parabola.(Loney XXX 18)

Prove that the normals at the points, where the straight line $lx+my=1$ meets the parabola, meet on the normal at the point $(\frac{4am^2}{l^2},\frac{4am}{l})$ of the parabola. As the line intersects the parabola we have $$y^2=4a \Bigg(…
mathemather
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The straight line $L_1$ touches the curves $y^2=4x$ at $A$ and $x^2=4y$ at $B$

The straight line $L_1$ touches the curves $y^2=4x$ at $A$ and $x^2=4y$ at $B$.The straight line $L_2$ is normal to both the curves cutting the first curve at $C$ and $D$ and the second curve at $E$ and $F$.Find the area bounded by $L_1$ and the…
learner_avid
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Challenging (or not) paralelogram problem - vector as a linear combination of other vectors

I'm having the hardest time on this particular problem: On the paralelogram ABCD we have $\vec{DE} = \alpha \ \vec{DC}$ $3 \ \vec{BF} = \vec{FC}$ $G$ is the intersection between $AE$ and $DF$ What is the value of $\alpha$ so that $\vec{AG} = \ 4…
bru1987
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Write the plane in vector equation form

$2x+4y-4z=4$ is the point normal equation form for the plane. How do we write the plane in vector equation form, as $$(x,y,z)=(*,*,*)+t_1(*,*,*)+t_2(*,*,*)$$
Guest
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Minimum area of a triangle

Question: A line is drawn through the point (1, 2) to meet the coordinate axis at P And Q such that OPQ is a triangle with O as the origin. If area of triangle OPQ is least, then what is the slope of line PQ. I have been given four options A(-1/4),…
Kartik Watwani
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Surface of cap from two intersecting spheres

We have two intersecting spheres $S_1$ and $S_2$ with radii $r_1$ and $r_2$. $S_1$ is centered at $(0,0,0)$, $S_2$ is centered at $(r_1, 0, 0)$. How do I calculate the surface of the intersection cap on $S_1$? I am looking for a formula of $r_1$ and…
Erik Kalkoken
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