Ellipse Definition and 406 Threads

Hello everyone, I am still relatively new to this site, so any mistakes I take full blame for. My question is: At what point(s) on the ellipse x^2+4y^2=4 is the slope of the tangent line 1/2sqrt3? I have found the derivative of the equation through implicit differentation (I came out with...

If (5,12) and (24,7) are the focii of a conic passing through the origin, then find the eccentricity of the conic Attempt: Found the centre as (h,k), midpoint of the given points. (x-h)^2/a^2+(y-k)^2/b^2=1 i put x=0 and y=0 as it passes through the origin. from the equation e^2=1-(b^2/a^2)...

Homework Statement What is thea are of the largest rectangle that can be inscribed in the ellipse 4x^2 +9y^2 = 36 A) 6 rad 2 B) 12 C)24 D) 24 rad 2 E) 36 Homework Equations Must be done using optimization and first derivitive The Attempt at a Solution I know I have to use A=...

Homework Statement Prove that the straight line x cos @ + y sin @ = p is a tangent to the ellipse x2/a2 + y2/b2 if a2 cos2@ + b2 sin2@ =p2 . u and v are the perpendicular distances of a tangent from the two points M(0,ae) and N(0,-ae) respectively. Prove that u2 + v2 is a constant...

Find Curvature of Ellipse given x=3*cos(t) and y=4*sin(t) at the points (3,0) and (0,4) Relevant equations: curvature at r(s) is k(s)=||dT/ds|| when r(s) is arc length parametrization and T is the unit tangent vector I usually use the formula k(t)= (||r'(t) x r''(t)||)/||r'(t)||^3 So...

Homework Statement an ellipse has the vertices (+&-4, 0)and the point (1,2) lies on the ellipse. find the standard form of the equation of the ellipse . Homework Equations (x-h)^2/a^2 + (y-k)/ b^2 = 1 The Attempt at a Solution (x-0)^2/(4)^2 + (y-0)/b^2 = 1 (1-0)^2/16 +...

An ellipse is a conic section. If you construct an ellipse using a cone, does the axis of the cone cross through one of the foci of the ellipse? if so, how can this be shown mathematically? This is just purely out of curiosity.

Homework Statement Determine the points on the ellipse x^2 a 2y^2=1 where the tangent line has a slope of 1 Homework Equations I'm able to solve problems when given points and asked to find equations of the tangent lines. However, I'm struggling to do the inverse. The Attempt at a Solution...

Homework Statement Use the discriminant to determine if the following are equations of an ellipse, parabola or hyperbola 6x^2-12xy+6y^2-5x+9=0 5xy-4y^2+8x-3y+20=0 x^2-9xy+5y^2-2=0 10x^2-9xy+5y^2-2=0 2y^2-10x+9y-8=0 Homework Equations The Attempt at a Solution I got these...

Homework Statement Find the equations of all the tangent lines to x^2 + 4y^2 = 36 that pass through the point (12,3) Homework Equations the derivative of the ellipse is dy/dx = -2x/8y (I'm not sure if that is correct, i have only recently learned implicit differentiation.) The...

Homework Statement The Earth travels in an elliptical path with the sun at one of its foci. Is the acceleration of the Earth directed towards the sun or towards the centre of the ellipse? Homework Equations The Earth's elliptical path can be parametrized as: x=acos (pt) y=bsin (pt)...

Why does the area of an ellipse have a closed form while the perimeter does not? Obviously if the area is finite so is the boundary so it seems the perimeter should be calculable in a closed form.

This is how the book describes the problem: If the ellipse x2/a2+y2/b2=1 is to enclose the circle x2+y2=2y, what values of a and b minimize the are of the ellipse? First of all I completed the square for the second equation and I got: x2+(y-1)2=1. I isolated the x2 and substituted it into...

Hi every body I have a bounce marks on a quistion that i know nothing about its about the Ellipse the Qustion is Find the standerd formula of the eelips which has foci on ( 1,1) and ( -1,-1) and it has a major axis with 4 units. i found that the center in at the origin...

Homework Statement Largest possible area of a rectangle inscribed in the ellipse (x2/a2)+(y2/b2)=1 Homework Equations Area of the rectangle = length*height The Attempt at a Solution I have it set up so that the four corners of the rectangle are at (x,y) (-x,y) (-x,-y) (x,-y) and that...

Homework Statement Rotating the ellipse x^2/a^2 + y^2/b^2 = 1 about the x-axis generates and ellipsoid. Compute its volume. Homework Equations The Attempt at a Solution

Homework Statement A particle moves around the ellipse ((y/3)^2)+((z/2)^2)=1 in the yz-plane in such a way that its position at time t is r(t)=(3cost)j+(2sint)k. Find the maximum and minimum values of |v| and |a|. (Hint: Find the extreme values of |v|^2 and |a|^2 first and take square roots...

the regular ellipse formula in 2D is x^2/a^2 + y^2/b^2 = 1. but how can it be transformed into a 3D formula including the parameter of z? thank you!

If I stretch an ellipse with .5 eccentricity along an axis 45 degrees from its major axis, doubling its area. How do I find the angle of the major axis of the resulting ellipse? Is there a simple rule based on the amount you stretch and the angle and orig eccentricity? Thanks.

Lets consider an ellipse with equation \frac{x^2}{25}+y^2=1 and a parabola with equation y=ax²+bx+c (on the same grid) meet at x = ±4 (y>0). This is in a way that at both points would have identical tangents, of course. In this situation, without making any graphs etc, how can someone...

I've been stuck with this problem: An ellipse can be defined as 1) locus of points for which is constant the sum of the distances from two fixed points (foci) 2) locus of points for which is constant the ratio between the distances from a fixed point (focus) and a fixed line (directrix)...

Problem: What is the area of the largest rectangle that can be inscribed in the ellipse 9x2+4y2=36 Relevant Equations: equation of an ellipse: (x - h)2/a2 + (y - k)2/b2 = 1 I only got as far as x2/4 + y2/9 = 1 (x - 0)2/22 + (y - 0)2/32 = 1 I have no idea where to go from here

Prove or disprove: The intersection of the plane x+y+z=1 and the cylinder x^2+y^2=1 is an ellipse.

Homework Statement Show that the area of x^2/a^2+y^2/b^2=1 is \pi ab Homework Equations Given transformations: x=au y=bv The Attempt at a Solution J(u,v) = a*b \int\int ((au)/a)^2+((bv)/b)^2 J(u,v) dudv \int\int u^2+v^2 J(u,v) dudv \int_0^{2\pi}\int_0^1 r^2 J(u,v)...

Homework Statement The equation of tangent is given t:2x+3y-2=0 and the equation of elipse E:x^2+4y^2=K Find "a" and "b" and the coordinates of touching point D. Homework Equations equation of elipse: b^2x^2+a^2y^2=a^2b^2 equation for touching: a^2k^2+b^2=n^2 equation for K (if...

Homework Statement A body is moving on a trajectory \frac{x^2}{a^2} + \frac{y^2}{b^2} =1 vith a constant speed v_{0} . Find its velocity \vec{v} and acceleration \vec{a} . Homework Equations As far as I know \vec{a} = \vec{a}_{\tau} + \vec{a}_{n} = \frac{dv}{dx}\vec{\tau} +...

The motivation behind my question stems from my own curiosity. There was recently a post in this forum titled "The Widest Point on an ellipse" (or something to that effect). In any event, I misread the title, as "The wildest". I got to thinking, and remembered from vector calculus there existed...

Homework Statement Find and verify parametric equations for an ellipse. Homework Equations x=acost y=bsint The Attempt at a Solution lets say the equation is x=3cost, y=3sint, domain: 0 to 2pi x2 y2 -- + -- = 1 a2 b2 point does verify when t=0 x=3, y=0 which =1...

Hello, given (x^2)/(a^2) + (y^2)/(b^2) = 1. and using polar coordinates x=rcos(phi) , y=rsin(phi), equating gives r^2 = 1/[(cos^2(phi)/a^2) + (sin^2(phi)/b^2)]. or if we leave b in the nominator : r= b/[(sin^2(phi)+(b^2/a^2)cos^2(phi)]^1/2. -could someone give a hint as to how the...

Actually, this is just a plain old geometry problem, really - no special tricks or anything. It just has a very nice solution, which someone showed me years ago (I take no credit). Can you prove that the intersection of a right circular cylinder and a plane is an ellipse? (Assume the...

Homework Statement Find the surface area obtained when the upper half of the ellipse: \frac{x^{2}}{4}+ y^{2}=1 is rotated about the x-axis Homework Equations \int2piyds The Attempt at a Solution

Given equation x^2/9 + y^2/4 = 1. Determine the two points on the ellipse having a tangent line passing through the point (0,3). Cant seem to figure this one out? I have found the derivative of the slope which is -4x/9y. I don't know how to use that slope in order to find a tangent line that...

Find the tangent lines to the ellipse x^2 + 7y^2 = 8 at the point (3,0) Slope-intercept form: y=mx+b I know you have to differentiate the equation implicitly to get the slope, but you come across a zero in the denominator and that has me stumped.

I'd like to map the open unit circle to the open ellipse x/A^2 + y/B^2 = 1. How would I go about doing this? I really have no idea how to go about doing these mappings. I'm working with the text Complex Var. and Applications by Ward and Churchill which has a table of mappings in the back...

Homework Statement (i) Evaluate \int_C \dfrac{-ydx + xdy}{9x^2 + 16y^2} when C is the ellipse \dfrac{x^2}{16} + \dfrac{y^2}{9} = 1 (ii) Use the ans to (i) to evaluate the integral along C' = ellipse: \dfrac{x^2}{25} + \dfrac{y^2}{16} = 1 Homework Equations The...

Sorry title was supposed to be Conic Sections, but my I key is sticky :) I had a question today, It went somethng like this: An epllipse of equation ((x^2)/4) + y^2 = 1 Find the equation of the tangent which passes through point P: (4,0) Well this was a mock exam question, where no answers...

parametrizing an ellipse -- what am I doing wrong? I have the width, height, and center point of the ellipse. I have an angle/vector from the center point and I want to know at what point it intersects the ellipse perimeter. Based on this wikipedia article: http://en.wikipedia.org/wiki/Ellipse...

Let P(x,a) and Q(-x,a) be two points on the upper half of the ellipse \frac{x^2}{100}+\frac{(y-5)^2}{25}=1 centered at (0,5). A triangle RST is formed by using the tangent lines to the ellipse at Q and P. Show that the area of the triangle is A(x)=-f'(x)[x-\frac{f(x)}{f'(x)}]^2...

Homework Statement x^2 - 2xy + 6y^2 = 10 Find the point on the ellipse closest to the origin (0,0).Homework Equations The Attempt at a Solution Absolutely no one in my class can solve this. We've been to the math lab and none of the helpers there know how to solve it. I think the only person...

Homework Statement Convert the conic section to standard form. r=\frac{1}{8-4*sin(\theta} Homework Equations x=rcos(\theta) y=rsinx(\theta) The Attempt at a Solution r=\frac{1}{8-4*sin(\theta} r^2=\frac{1}{64-64*sin(\theta)+16sin^2(\theta)} r^2= x^2 + y^2 I can see the...

The question I am looking at states: Prove that the elliptical function x^2 + 3y^2=b is orthogonal to the cubic y=3ax^3. I'm thinking they want me to prove if the functions are orthogonal when they intersect. If I am correct that would just lead to; For the ellipse, 2x+6y(m1)=0...

Homework Statement The graph of the tilted ellipse x^2 -xy +y^2 =3 is shown to the right. What are the dimensions and the location of the box containing the ellipse? Note the sides of the box are vertical and horizontal and also are tangent to the elipse. (The image is simply a tilted...

Homework Statement The problem is that an ellipse (centered at origin) is revolved about y-axis. Now I have to find the volume of this swept region. But how do I go about using calculus? I have to derive it. Homework Equations Volume of ellipsoid = 4/3*pi*abc (source wikipedia) Equation...

Homework Statement How would you derive the equation for an ellipse from the parametrization: x = a cos(t) y= b sin(t) If I solve for t and set them equal, I get: arccos(x/a) = arcsin(x/a) which looks nothing like the usual formula: x^2/a^2 + y^2/b^2 = 1 ? Homework...

Homework Statement an ellipse whose semi axes have lengths a and b rolls without slipping on the curve y =c sin (x/a), find the relationship between a, b, and c. Assume that the ellipse completes one revolution per period of the sine curve. The answer is b^2 = a^2 + c^2 and you find it by...

Homework Statement I was tutoring a student and could not answer one of his questions. Prove that y = mx +- square root( a^2*m^2 + b^2 ) is the equation for two lines passing through an ellipse Homework Equations (x/a)^2 + (y/b)^2 = 1 is the equation of an ellipse The Attempt at a...

Find the equation of a ellipse given the foci. (1,0) (3,4)

Hello all In radio astronomy the orientation angle of an ellipse is usually quoted as tan(2 *Psi) = S2/S1 Where S2 and S1 are the stokes parameters. Does anyone know or can point me to a reference as to why the 2 * psi should be in there? The angle of the axis in the ellipse should be...

I'm revising form my A-levels now and I ran into a bit of problem with a question. It looks easy, but I can't get the answer at the back of the book. Could be a typo, but could be me that's wrong. Question: The eccentric angle corresponding to the point (2, 1) on the ellipse with equation x^2...

How do you find the total width of the ellipse given by the equation 7x^2 + 7(y-6)^2 = 6?