What is Coordinate geometry: Definition and 56 Discussions
In classical mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry using a coordinate system. This contrasts with synthetic geometry.
Analytic geometry is used in physics and engineering, and also in aviation, rocketry, space science, and spaceflight. It is the foundation of most modern fields of geometry, including algebraic, differential, discrete and computational geometry.
Usually the Cartesian coordinate system is applied to manipulate equations for planes, straight lines, and squares, often in two and sometimes three dimensions. Geometrically, one studies the Euclidean plane (two dimensions) and Euclidean space (three dimensions). As taught in school books, analytic geometry can be explained more simply: it is concerned with defining and representing geometrical shapes in a numerical way and extracting numerical information from shapes' numerical definitions and representations. That the algebra of the real numbers can be employed to yield results about the linear continuum of geometry relies on the Cantor–Dedekind axiom.
The equation of ellipse reduces to :
$$(2x+3)^2+(3y+2)^2=8$$
$$\frac{(x+3/2)^2}{8/4}+\frac{(y+2/3)^2}{8/9}=1$$
Center of ellipse =##\left(\frac{-3}{2},\frac{-2}{3}\right)##
##b^2=a^2(1-e^2)=8/9## and ##a^2=8/4##
Therefore ##e=\frac{\sqrt{5}}{3}##
Distance between foci=##\frac{2\sqrt{10}}{3}##...
So I tried to solve it this way:
The equations of the lines in vector form are
$$(x-2)\hat i+(y-3)\hat j+(z-4)\hat k=\lambda (\hat i+\hat j-K\hat k)$$
$$(x-1)\hat i+(y-4)\hat j+(z-5)\hat k=\mu (K\hat i+2\hat j+1\hat k)$$
Since the lines are some real multiple of the vectors,
For coplanarity...
About two years ago I had a thread here, where I was looking for a way to calculate the distance between two points A and B on a plane when the following information is given:
The distances of the points A and B to three arbitrary (but non-colinear) reference points.
The distances between the...
The solution in my book is as follows:
"Since the line has equal intercepts on axes, it is equally inclined to axes.
\implies line is along the vector a(\hat i + \hat j + \hat k)
\implies Equation of line is \frac{x-2}{1}=\frac{y-3}{1}=\frac{z-4}{1}"
As per my understanding, an intercept is the...
Heres how I proceeded,
Equation of line ##AC## in vector form:
$$\vec r=a+t(c-a)$$$$\vec r=(1i+4j+3k)+t(2i-6j+2k)$$
Since ##B## doesn't lie on ##AC## ##b\neq (1+2t)i +(4-6t)j+(3+2t)k##
The following equation is derived:
$$2\hat i+\alpha \hat j+4\hat k\neq (1+2t)\hat i +(4-6t)\hat j+(3+2t)\hat k...
I'm trying to review some high school maths and work my way to Calculus and Linear Algebra, and I found these three translations of Japanese maths textbooks translated by the AMS and edited by Kunihiko Kodaira. The AMS links to them are:
https://bookstore.ams.org/cdn-1669378252560/mawrld-8/...
I have come across Cartesian, Polar, Cylindrical & Spherical Coordinate Systems so far and was wondering if someone could tell me which are the uncommon systems used in physics which everyone says that they exist but no one explicitly mentions. Is there a "standard reference" or are they just...
Summary:: The set of values of ##b## for which the origin and the point ##(1, 1)## lie on the same side of the straight line ##a^2x+aby+1=0## ##\forall~a\in\mathbb{R},~b>0##.(a) ##a\geq1## or ##a\leq-3##
(b) ##a\in~(-3,~0)\cup(\frac13,~1)##
(c) ##a\in~(0,~1)##
(d) ##a\in~(-\infty,~0)##
I tried...
1.To shift the graph of a function :
Vertical Shifts : ## y=f(x) +h## where the graph shifts ##k## units up if ##k## is positive and downwards when ##k## is negative.
Horizontal Shifts : ##y=f(x+h)## where the graph shifts to the left by ##h## units when positive and to the right when ##h## is...
The problem :
A variable line, drawn through the point of intersection of the straight lines ##(x/a)+(y/b) = 1## and ##(x/b)+(y/a)=1##, meets the coordinate axes in A & B . We have to Show that the locus of the mid point of AB is the curve ##2xy(a + b) = ab(x + y)##.
The solution :
Let ##(h...
Given the coordinates ##P = (3,4)## , find the coordinates of ##P"(x',y')## when the origin is shifted to (1, –2), and the axes are rotated by 90° in the clockwise direction.
I attempted to solve this problem using the following formulas :
##x = X + h## and ##y = Y + k## for translation of the...
Hi, I am really stuck! I need to find the equation of the plane through the line x=2y=3z perpendicular to the plan 5x+4y-3z=8. Can anyone give me any pointers of where to start with this? Not expecting a full solution, just an idea of where to start.
THanks!
Homework Statement
The line "4x + 3y -4=0 " divides the circumference of circle centred at (5,3) in ratio 1:2.whats the equation of the circle
Homework Equations
2πr
Distance of a line from point
Radius of general circle
The Attempt at a Solution
I tried finding the distance of the centre...
Hello
In India, SL Loney's Elements of Coordinate Geometry is very popular for entrance examinations. I wanted to refresh my coordinate geometry, so tried reading through the book. But I found that the language used is old. I found myself referring to current material on the topic to properly...
Homework Statement
If G be the centroid of ΔABC and O be any other point, prove that ,
## 3(GA^2 + GB^2 + GC^2)=BC^2+CA^2+AB^2##
##and,##
##OA^2 + OB^2 + OC^2 = GA^2.GB^2+GC^2+3GO^2##
Homework Equations
i m practising from S L LONEY coordinate geometry first chapter ... only the equation...
Homework Statement
If the normal at P(ap^2 ,2ap) to the parabola y^2 = 4ax meets the curve again at Q(aq^2, 2aq), show that p^2 +pq+2=0
Homework Equations
Point-slope form
The Attempt at a Solution
I tried putting y=2aq and x=aq^2 but I can seem to simplify the whole thing other than...
.The number of points, having both co-ordinates as integers, that lie in the interior of the triangle with vertices (0, 0), (0, 41) and (41, 0), is
(1) 901 (2) 861 (3) 820 (4) 780
my attempt:
for this to be true i know that sum of x and y coordinate should be 41 but i don't know how to proceed.
Homework Statement
If two vertex of a square of the side AB is A(1,2) and B(2,4) find other two vertex C and D?
Homework Equations
1. y-y1 = m ( x-x1)
2. m=y1-y2 / x1-x2
3. m1*m2= -1
4. (x-x1) / (x1-x2) =(y-y1)/(y1-y2)
5. m=tanA
6. If ax+by+c=0 then its parallel line is ax+by+k=0
The Attempt...
Find the equation of the ellipse whose eccentricity is 2/3 and which has (2,0) and x+y=0 for focus and corresponding directrix .
given answer: (x-2)^2 + y^2 = 4/9 ( ( (x+y)/√2)^2)what i tried doing:-
ae=2
⇒a*2/3=2
∴a=3
found b=√5.
what to do next? please help.
< Mentor Note -- thread moved to HH from the technical math forums, so no HH Template is shown >
Find the equation of a line passing through point A (1, 2) and whose perpendicular distance from origin is maximum.
Homework Statement
if the tangent at a point P("theta") on the ellipse
16 (x^2) + 11 (y^2) = 256
is also tangent to the circle
(x^2) + (y^2) + 2(x) = 15
then ("theta") = ??
2. The attempt at a solution
{{{{ i have taken "theta" as "d" }}}}
P [4 cos d , (16/(sqrt11)) sin d]
equation of...
Definition/Summary
In Euclidean coordinate geometry, a Line usually means the whole (infinitely long) Line.
(In ordinary Euclidean geometry, a Line usually means a line segment between two points.)
The equation of a Line in n dimensions is a combination of n-1 linear equations of the...
Let A(1, 2), B (3,4), C( x, y) be points such that (x- 1) (x-3) +(y-2) (y-4)=0. Area of triangle ABC=1.
maximum number of positions of C in the xy plane is
(a) 2
(b) 4
(c) 8
(d) None of these
I have tried using the staircase formula which gives me something like x-y=2. Therefore I see only...
Hello,
Please forgive me if it's in the wrong sub-forum because don't know where to place it.
I need help solving this problem it's chapter 1 and in our class we are already in chapter 5 so I might sound like a fool asking the teacher about it, I was revising and decided to do some questions...
Author: H.M. Schey
Title: Div, Grad, Curl, and All That: An Informal Text on Vector Calculus
Amazon Link: https://www.amazon.com/dp/0393925161/?tag=pfamazon01-20
Prerequisities: Calculus 1,2,3
Table of Contents:
Preface
Introduction, Vector Functions, and Electrostatics
Introduction...
I would like someone to give this a quick check, I am really not sure if I am over thinking this question. I got the right ans, just would like a quick check of my method; big thanks in advance.
question: P(-1,5), Q(8,10), R(7,5) & S(x,y) are the veritices of the parallelogram PQRS. Calculate...
find the equation of a circle whose center falls ont he line y=6-2x and which passes through the points A(-2,0) and B(4,0).
poor in circles. how to even start?
Dear all,
I am a medical doctor stuck with a mathematics problem. I have taken a project to map the 3D movement of a particular bone and describe its motion. My maths knowledge is a bit lacking. However using logic I felt that I would need to track 3 constant markers on the particular bone...
Homework Statement
Given that P,Q and R lie on the hyperbola xy=c2, prove that if PQ and PR inclined equally to the coordinate axes, then QR passes through O.
Homework Equations
The Attempt at a Solution
I don't understand what does ''PQ and PR are inclined equally to the...
Homework Statement
2 circles have the equation x2+y2-2x-2y+1=0 and x2+y2-12x-12y+36=0 respectively. Both circle touches the x-axis, y-axis and the line 3x + 4y = 12. Find the fourth tangent of the 2 circles.
Homework Equations
The Attempt at a Solution
This is second part of the...
Homework Statement
A variable point P lies on the curve y2 = x3 and is joined to a fixed point A with coordinate (2,0). Prove that the locus of the mid-point of AP is y2= 2(x-1)3.
Homework Equations
The Attempt at a Solution
According to what i know, I need to know the...
if the 3 spatial dimensions are represented on a 3 axis graph. would time be a straight line through the origin? if not could your direct me to some reading that i may find answers? I'm uncertain whether space-time can be represented using euclidean geometry.
Hi
Homework Statement
Two points X(a,b) and Y(b,a)
Prove that the distance = sqrt2 (a-b)
Homework Equations
d = sqrt(x2-x1)^2 + (y2-y1)^2)
The Attempt at a Solution
I have drilled it down to
(sqrt 2b^2+2a^2-4ba) but unable to drill it down to sqrt2 (a-b)
Help...
Homework Statement
The points A and B are located on the line 5x-12y=6.The perpendicular distances of A and B respectively from the line 3x-4y=-2 are both 4 units. A lies on the same side of 3x-4y=-2 as the origin ,and B lies on the other side. Find the coordinates of A and B.
Homework...
Homework Statement
Using the concept of equal slopes, state whether the following sets of points are collinear or not
A(5, -6) B(0, -1) C(-4, 3)Homework Equations
Slope = vertical rise/horizontal runThe Attempt at a Solution
I only got as far as...
5-0/-6+1 = 5/-5
I think what I just did...
Homework Statement
Fairly straight forward question but I just can't see what's wrong.
A circle passes through the point (2,-4) and touches both the x-axis and the y-axis. Find the equations of the two circles which satisfy these conditions.
Homework Equations
x^2 + y^2 + 2gx + 2fy...
Homework Statement
Suppose that II c R3 is a plane, and that P is a point not on II. Assume that Q is a point in II whose distance to P is minimal; in other words, the distance from P to Q is less than or equal to the distance from P to any other point in II. Show that the vector PQ is...
is one -one correspondence must for a theory like coordinate geometry , polar coordinate ,vector analysis etc
to work , i.e theories which work by representing a quantity by a different set of quantities
behaving alike
Help neede in coordinate geometry!
Homework Statement
here is the question :
If the pairs of lines x^2-2pxy-y^2 and x^2-2qxy-y^2 are such that each pair bisects the angle between the other pair , then pq equals ...??the answer somehow is -1. i think it uses the concept of homogenisation (if u...
Question Statement
The straight line y=mx -2 intersects the curve y^2=4x at the two points P(x1,y1) and Q(x2,y2). Show that
1) m>-1/2, m not equal to 0.
2) x1 + x2 = 4(m+1)/m^2
3) y1 + y2 = 4/m
If the point O is the origin and the point T is a point such that OPTQ is a parallelogram, show...
The points Q moves such that the length of the tangent from Q to the circle
x^2+y^2+4x+8y+9=0 is equal to the distance of Q from the origin ). Determine the locus of Q.
I am basically clueless about this question...but I will try to provide as many work I have done on this question.
I assume...
The points P(cp,c/p) and Q(cq,c/q) lie on the rectangular hyperbola xy=c^2. Show that the equation of the chord PQ is pqy+x=c(p=q), and deduce the equation of the tangent at the point P.
I can do the proving. And I can find the tangent as well if the word "deduce" is not there. How can you...
Homework Statement
Sketch the curves with the following equations:
a) y=5x^(3/4)
b) y=-2x^(-3/2)
Homework Equations
The general equation of curves (circles, parabola, ellipse, hyperbola)
The Attempt at a Solution
It seems to me that this question is not exactly a coordinate geometry...
Not actually homework, but just a general query...
I'm trying to write a program that will predict the trajectory of a moving point by analysing several positions through which the point has passed.
Part of this involves finding the coordiantes of a point B, given the coordinates of another...
Hi all (<--new user) I've spent the last 6 months trying to learn A-level mathematics in a vacuum after 9 years of letting my GCSE knowledge slide into oblivion, this is the first time however, I've hit a question for which how the book arrives at its answer is completely frying my brain. The...
Hi there, I've been struggling with this question for days :cry: :confused: , the first part where they call me to prove that equation, i could do it...the second one i don't know how to do...could anyone help me? how do i find the locus equation? its so confusing... ty in advance.
If...
The question is in the attached picture.
i can't seem to find a relevant formula and when i do i can't get a suitable answer, if some one can help, i'll highly appreciate it. Thanks in advance.