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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How to identify any point inside or outside the given cone?

The equation of a double circular cone with a vertex $p=(a,b,c)$ with the generating angle $t$ is given by $(x-a)^2+(y-b)^2= \frac{(z-c)^2}{t^2}$ How do I identify the point ...
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How to find centre,vertics,foci,focal radii,letus rectum… when exists of a general quadratic equation in x and y

Is there a generalized way( a particular conic section of any shape,for instance an ellipse without determining its major/minor axis) to find the centre,vertics,focus,focal radii,letus ...
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27 views

How do you sketch a parabola given the equation? [on hold]

The equation is $y^2=16x$, and it wants you to sketch the vertex, focus, directrix.
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need explanation of what exactly is a directrix & focus?

((I'm not asking why do we need to know conic sections etc.) Like other similar questions.) I actually love math & currently learning conic sections in class, neither my textbook or teacher ...
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2answers
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Area of triangle inscribed in a parabola

How can u prove that the area of the triangle inscribed in a parabola is twice the area of the triangle formed by the tangents at the vertices?
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Centroid of triangle formed by co-normal points

How can you prove that he centroid of a triangle formed by 3 co-normal points lies on the axis of the parabola?
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Orthogonal tangents to an ellipse

This is the problem I found back in the first year in the university. Suppose we have a non-degenerate (i.e. not a point and not an empty set) ellipse $E\subset \Bbb R^2$. Now define a set $D$ by a ...
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The minimum distance from the circle $x^2+(y+6)^2=1$ to parabola $y^2=8x$?

What are the coordinates of the points on the parabola $y^2=8x$ which are at the minimum distance from the circle $x^2 + (y+6)^2=1$?
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Counting the dimension of a component of $\mathsf{hilb}^{2t+1}_{3}$

Consider the Hilbert scheme $\mathsf{hilb}^{2t+1}_{3}$, parametrizing varieties of degree $2$ and genus $0$ in $\mathbb{P}^{3}_{k}$, with $k$ an algebraically closed field. Consider the component $ ...
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Locate a point a given distance from another point on an ellipse

Similar to Point on circumference a given distance from another point, but for an ellipse. Unfortunately, the difference is non-trivial. I have an ellipse and a point (C) that is somewhere on the ...
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how do i write an equation in standard form by completing the square for $x^2 -9y^2-4x-18y=14$

I'm really having trouble with completeing squares i can solve for circles and ellipses but i can't seem to understand hyperbolas or parabolas, help would be deeply appreciated.
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Calculating an Ellipse given the Orbital Eccentricity and a Vertex?

I know that the formula for Eccentricity is e = c/a where c is the distance from the center to a focus and a is the distance from that focus to a vertex. I know the distance from the center of the ...
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27 views

Finding locus of centroid

Let AB be a chord of circle x^2 + y^2 = 3 which subtends 45 angle at P where P is any moving point on the circle. Then find the locus of centroid of triangle PAB Any help would be appreciated
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1answer
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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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1answer
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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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Hyperbolas defined by three points

Let's take three non-collinear points $A,B,C$. Take a hyperbola with $A,B$ as foci passing through $C$, a hyperbola with $B,C$ as foci passing through $a$ and a hyperbola with $C,A$ as foci passing ...
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35 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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1answer
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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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1answer
31 views

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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2answers
51 views

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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1answer
56 views

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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3answers
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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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1answer
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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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3answers
215 views

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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finding the center of an ellipse given 3 points and 2 tangent lines

I'm given a vertex on the minor axis- point (0,7), two points on the ellipse, (-15,0) and (15,0) and the tangent line in point (-15,0) has an angle of 65° to the x axis (so actually two tangent ...
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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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2answers
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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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Parabolic regression with restricted shape

How can I calculate the parabolic regression with vertex at minimum. Is it possible? I have a set of points from which I estimate the parabola using the (I believe) standard equation (from ...
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1answer
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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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1answer
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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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1answer
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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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1answer
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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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1answer
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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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1answer
26 views

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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1answer
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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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1answer
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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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conics ellipse and circle

If $a$ and $c$ are positive real numbers and the ellipse $\frac{x^2}{4c^2} + \frac{y^2}{c^2} =1$ has four distinct points in common wt the circle $x^2+y^2=9a^2$ then a) $9ac-9a^2-2c^2 < 0$ b) ...
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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 ...