Questions on conic sections and their properties; the curves formed by the intersection of a plane and a cone. Circles, ellipses, hyperbolas, and parabolas are examples of conic sections.

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Finding Radical centre problem

Suppose 3 circles are drawn taking the 3 sides of a triangle as their diameters, what would be the radical centre of these circles? The options are circumcenter, orthocenter and incenter Any help ...
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Locus of centre of variable circle

I am not able to figure out this question What is the locus of the centre of a circle which touches a given line and passes through a given point, not lying on the given line? I think it's a ...
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51 views

Intersection of a 45 degree angle and an ellipse

If you are looking at the upper right quadrant of an ellipse centered at $(0,0)$, with $a=1$ and $b = 0.6$, and there is a $45$ degree line drawn from $(1, 0.6)$, how would I find the $(x,y)$ ...
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61 views

Identify the locus.

Let $A,B,C$ lie on a straight line. $B$ is lying between $A$ and $C$. Consider all circles passing through $B$ and $C$. The point of contact of the tangents from $A$ to these circles lies on ..... We ...
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If an ellipse has two radiuses, is there something like it, but with three or more radiuses?

If we say that a circle has one radius, and an ellipse has two, can I define figures that have three, four, or more radiuses? Also, how can I get that "radius"? In an ellipse that is 10 at its ...
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How to find the equation of a parabola with vertex on the line y = -3x?

Its axis are parallel to the y-axis and passing through (-7,13) and (5,1).
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Find the tangents to the following curve from the given point.

2x^2 + y^2 = 54 from (10,1) P.S. I still don't study calculus. This lesson is from analytic geometry and I have no idea how to solve it because my professor didn't teach it. So if someone could tell ...
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Equation of intersection of two cones

The equations of two cones are given; $(x-x_{0})^2+(y-y_{0})^2=\frac {(z-z_{0})^2}{m^2}$ and $(x-x_{1})^2+(y-y_{1})^2=\frac {(z-z_{1})^2}{m^2}$ How to find the equations of intersections 1) ...
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Find the equation of an ellipse

I have to find the equation of an ellipse which passes through the point $(3, 2)$, has center at the origin and major axis along the y-axis, i.e., is a vertical ellipse. No other info is given. I've ...
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57 views

Solve a system of equations involving two ellipses

Problem #38 asks us to solve the system using either graphing, substitution, or elimination. The only way that I can think of doing this is by graphing. However, is there any easy way to solve this ...
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82 views

Area of circle formed when sphere is sliced by a plane

First off, when a sphere is cut by a plane, is a circle always formed or does a ellipse get formed in some cases? If a circle is always formed, how do you prove it? Next, how would you find the area ...
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Why does the “T=0” method to calculate tangent work?

Given a random equation of a curve: $ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0$. Suppose we need to find the tangent to this curve at any point $A(x_1, y_1)$. A method given to me by my professor was the ...
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Show that the intersection of a plane…

Show that the intersection of the plane $z = 2y$ with the elliptic cylinder $\frac{x^2}{5} + y^2 = 1$ is a circle. Find the radius and center of this circle. Hint: How can one describe a circle in ...
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Number of ellipses to uniquely define a co-centered circumscribing ellipse

I have a bit of a tricky problem that has come up in my engineering research, but I haven't quite got the brains to figure it out, though I've gotten pretty far. Suppose that there is an unknown ...
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What is wrong with this method for a rotated and shifted parabola?

$(x+2y)^2=4(x-y)$ Disecting the above parabola is the question. (vertex, axis,tangent at vertex,etc). So at first what I thought of was making its equations at LHS and RHS perpendicular. I thought ...
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Graphing an ellipse on TI-nspire CX CAS

How do I graph an ellipse on a TI-nspire CX CAS? I know how to graph an ellipse with the equation $$\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2 }=1$$ but I don't know how to put coefficients in the ...
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Conic Sections - Why do I need to know all these terms (foci, latus rectum, directrix, etc)? When will I use them?

I believe in learning something because I want to. If I do not want to learn about a subject or concept, I will not learn it well and master it. I am currently learning about conic sections, and I am ...
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Quadric question

I'm trying to prove that given 3 disjoint lines in $\mathbb{P}^{3}$ there exists a non-singular quadric containing them. The exercise is from the following link: ...
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62 views

3D Graphing--finding an equation given a graph

I'm having trouble finding a reasonable equation for this graph: http://i58.tinypic.com/15gtrn7.png The x axis is the horizontal, y-axis is the axis coming out of the screen, the z-axis is vertical. ...
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58 views

How do you find an equation for a locus?

Part 1 Given a directrix at x=-8 and a focus point at (-2,0), what are 5 points where the distance to the directrix is twice as far as the distance to the focus? Example: (4,0) is one of the 5 ...
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Regular division of the perimeter of an ellipse

I would like to divide an ellipse into $N$ parts such that these $N$ parts have the same arc length. So given let's say $a$ and $b$ the semi-axis of an ellipse centered on $(0,0)$ and a positive ...
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Parabolic word problem

A rectangular barge is traveling under a bridge with a parabolic archway. The barge is 60 feet tall and 80 feet wide. The bridge is 80 feet tall and 200 feet wide. If the barge must travel down the ...
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Finding a hyperbola's equation based off given asymptotes

I need help finding the equation of a hyperbola that opens vertically with asymptotes $y=2x+11$ and $y=-2x-1$. I also need help finding the equation of a different hyperbola that also opens upwards ...
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Longest parallel chord of an ellipse

I am searching for a source demonstrating that, for any set of parallel chords spanning an ellipse, the longest chord passes through the center of the ellipse. I am not referring to the major and ...
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Moving between different ellipse representations

I have a representation of an ellipse that is the affine transform of the unit ball, $\|Ax + b\| <= 1$. My question is, how can I change this ellipse representation? I would like to have it in ...
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What software should I use to graph this? / How do I get rearrange this equation so that it is in terms of y?

I thought I'd just quickly tell you guys why I want to graph this equation before giving it you. We're studying conic sections at the moment, and I started wondering what would happen if I let the ...
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How do I find the width of a given section of an ellipse?

How would I be able to find the width of a horizontal ellipse (with a major axis of 120 and a minor axis of 5) at any given point along the major axis?
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64 views

Find the arc-length of the circle with radius a?

Find the arc-length of a circle with radius a. From the equation of a circle, I found out the equation for the one quadrant, which is: $y = \sqrt{a^2 - x^2}$ I tried solving the problem, and here's ...
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Conic equation from cone/plane intersection

In an orthonormal cartesian frame $(O; \vec{x}, \vec{y}, \vec{z})$ consider: an infinite plane $P$ defined by: a point $p = (p_x, p_y, pz)$ an normal vector $\vec{n} = (n_x, n_y, n_z)$ a cone $C$ ...
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What are the coordinates of the ends of the latus rectum of the parabola $x^2 - 2y + 2 = 0$? [duplicate]

I've already graphed the parabola . i just don't know how to locate it's focus and the ends of it's latus rectum. On my graph, the vertex is on (0,1). Please help me with this. ASAP.
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Show that endpoint of a focal chord is $$\left(\frac{4p^2}{x_0}, \frac{p^2}{y_0}\right)$$

If $PQ$ is a focal chord of the parabola $x^2=4py$ and the coordinates of $P$ are $(x_{0}, y_{0})$ show that the coordinates of $Q$ are $$\left(\frac{4p^2}{x_0}, \frac{p^2}{y_0}\right)$$ I labeled ...
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Minimum eccentricity of ellipses around another ellipse

Six circles can surround another circle of equal size, with each circle touching both the central circle and its two neighbouring outer circles. For sufficiently eccentric ellipses, it is possible to ...
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ratio of tangent to the ellipse

The tangent at point $P = ( a \cos \phi, b \sin \phi)$ on the ellipse $\frac{x^2} {a^2} + \frac{y^2}{b^2}=1$ meets the $x$ and $y$ axes at the points $X$ and $Y$, respectively. Find in terms of ...
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Finding the maximum and minimum values on ellipse [closed]

Find the maximum and minimum values of f(x, y) = 5x + y on the ellipse x^{2} + 4y^{2} = 1
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Ellipse Word Problem

The ellipse is 5 meters across and 8 meters long with decorative fountains located at the foci. How far from the center should the fountains be located? (Rounded to the nearest hundredth). How far ...
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Find the equation of a circle passing three points (conics)

Problem: Determine the equation of the circle that passes through three points, $J(-3, 2)$, $K(4, 1)$, and $L(6, 5)$. I thought of using systems like so: $$\left\{ \begin{array}{rcl} (x+3)^2 + ...
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Conic Sections Question - Hyperbolas & Circles

So, if you have a hyperbola with foci at $(4,0)$ & $(-2,0)$, and the slopes of the asymptotes are $+4$ and $-4$, what would the equation for this hyperbola be? I know that the center would be ...
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How do I show that the equation E(k) = 2-4cos(ka) is a parabola when k=0 and when k=pi/a?

It's evident from the graph but I'm not sure how to show this mathematically. This dispersion relation is supposed to be roughly parabolic
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Conics: Asymmetric Hyperbola

I'm sure we've all seen the image below that illustrates the creation of the four conic sections. Although I've seen this multiple times throughout my education, I find it odd that the following case ...
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Intersection between conic and line in homogeneous space

In homogeneous space (so 3 coordinates for each point) I have: A conic C, defined by a symmetric 3x3 matrix of real values. The conic actually should have only imaginary points (don't know if this ...
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any conic in $\mathbb{A}^2$

Exercise 3.1 in Hartshorne's Algebraic Geometry: Show that any conic in $\mathbb{A}^2$ is isomorphic to $\mathbb{A}^1$ or $\mathbb{A}^1-\{0\}$. when the conic given by $x^{2}+y^{2}-1$, what the ...
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Non-linear systems help!

I have a non-linear system of equations, $$\left\{ \begin{array}{rcl} x^2 - xy + 8 = 0 \\ x^2 - 8x + y = 0 \\ \end{array} \right.$$ I have tried equating the expressions (because both equal 0), which ...
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Show that lines created by certain points on the parabola intersect at the directrix?

Edit: I got the answer by finding points of intersection between the line passing through B and the focus and the parabola, but it didn't seem like the best solution. Any other ideas? The Segments ...
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Find points of intersection with cone on a plane at a given angles

The provided variables are the cone angle(cA) of a cone that starts at the origin along the Z axis, the vertical angle (vA) of the direction the cone is facing, and a horizontal angle (hA) along with ...
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How to compute characteristic polynominal of two conics

If I have two conics defined as $A: XAX^T$ and $B: XBX^T$ how can I expand characteristic polynomial $f(\lambda) = det(\lambda A + B) $ so that it can be computed by a computer program or Matlab?
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How to find the height ($z$) on an elliptic cone at a point $(x, y)$

I am attempting to write a java method which returns the height of an elliptic cone given a $(x, y)$ point within the base. I have an elliptic cone centred at $(x_1, y_1)$, the major axis a, minor ...
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Finding a positive definite matrix to satisfy the general equation of an ellipse

I am trying to find a matrix A such that $(1)$ can be written as $v^TAv=1$ where $v=(x, y)^T$. $(1)$: $$\left(\frac{x}{a_1}\right)^2 + \left(\frac{y}{a_2}\right)^2 - ...
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Problem in conics question

A vertical line passing through the point ($h$,0) intersects the ellipse $$\frac{x^2}{4}+\frac{y^2}{3}=1$$ at the points P & Q.Let the tangents to ellipse at P & Q meet at the point R.If ...
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Area inside an ellipse

Given the ellipse x^2/25\ + y^2/16 = 1, A = (5,0), B = (0,4); Find point C (with both coordinates positive) on the ellipse, such that the area between AC and the ellipse (S1) will be equal to the area ...
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Is this a correct way to derive the equation of an ellipse/hyperbola?

I was just testing to see if I could derive the equation of an ellipse (and consequently a hyperbola) with the least amount of information to remember. The small amount of information I chose to use ...