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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Foci of ellipse and distance c from center question?

I don't understand how you would figure out an exact formula for the linear eccentricity (distance from the center to either focus) $c$ of an ellipse, being $c^2=a^2-b^2$, where $a$ is the length of ...
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34 views

Equation of the locus of the foot of perpendicular from any focus upon any tangent to the ellipse ${x^2\over a^2}+{y^2\over b^2}=1$

Find the equation of the locus of the foot of perpendicular from any focus upon any tangent to the ellipse ${x^2\over a^2}+{y^2\over b^2}=1$. will it also be an ellipse?
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74 views

Intersection between sphere and ellipsoid

I am failing since two days to compute and to plot the intersection of an ellipsoid in parametric notation ...
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22 views

Cubic curves vs conics

What is the main difference between cubic curves and conics, i.e. why can cubic curves develop singularies while conics cannot? Is this in some way related to Poincare-Bendixon theorem of chaos ...
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1answer
59 views

Which conic is represented by $r = a \cos \theta$

The polar equation $r = a \cos \theta$ represents which conic?
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32 views

Problem on co-ordinate geometry

Suppose the circle with equation $x^2 + y^2 + 2fx + 2gy + c = 0$ cuts the parabola $y^2 = 4ax$, ($a > 0$) at four distinct points. If d denotes the sum of ordinates of these four points, then find ...
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1answer
32 views

Finding the focus point of a conic with equation $ay^2 + bx = 0$.

A conic has equation $$ay^2+bx=0$$ where $a=5$ and $b=-315$. If the focus point is at $(F, 0)$ then what is the value of $F$ to 2 decimal places? Hi, I want to check if i have applied the correct ...
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1answer
40 views

Equation for focus and directrix

Is it possible to get a focus and directrix straight from the equation itself or through a formula? For example, in $y = (x-2)^2 + 1$, you can tell from the equation that the vertex is $(2,1)$. Or ...
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2answers
52 views

Hyperbolas - Standard Form

This is probably a simple question but if $y = \frac{1}{x}$ is a hyperbola, then how does it comply with the standard form of a hyperbola?
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1answer
63 views

ellipse circumference

Here is a Wikipedia article about the circumference of an ellipse: http://en.wikipedia.org/wiki/Ellipse#Circumference I don't know how Ramanujan developed the following approximation for the ...
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2answers
106 views

Ellipse Diagonal's Length/Equation [closed]

Excuse the vagueness of this question, but how can you find the equation and distance for the diagonal of any given ellipse, that is, the line from the most-northwestern point to the most southeastern ...
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3answers
151 views

Focus of parabola with two tangents

A parabola touches x-axis at $(1,0)$ and $y=x$ at $(1,1)$. Find its focus. My attempt : All I can say is that as angle subtended by this chord at focus is $90^\circ$ as angle between tangents is ...
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1answer
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Rotation of conics sections using linear algebra

When given an equation of the form $$Ax^2+Bxy+Cy^2 + Dx + Ey + F$$ where $B \not= 0$ and it is not a degenerate conic, then you can use $\Delta = B^2 -4AC $ to see what type of conic it is, and then ...
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1answer
36 views

Finding perimeter of an ellipse accurately

How could you accurately find the perimeter of an ellipse accurately? This formula: $$p\approx 2\pi\sqrt{\dfrac{a^2+b^2}{2}}$$ (Where 'a' is the distance from the center of the ellipse to the ...
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4answers
73 views

Interesting association between tangent lines of slope one and ellipses

Why is it that a tangent line with slope $1$ to an ellipse centered at the origin will have a transformation of $\pm \sqrt{a^2 +b^2}$ where $a$ and $b$ are the major and minor axis of the ellipse? ...
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0answers
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Finding the distance from a parabola (ballistic trajectory) to a point (for use in collision detection)

I need to have some form of collision detection / prevention for an object moving along a ballistic trajectory and a second stationary object on the same plane plane. The ballistic trajectory is ...
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1answer
27 views

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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1answer
64 views

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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1answer
30 views

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
90 views

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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1answer
38 views

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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2answers
64 views

Orthogonal tangents to an ellipse [duplicate]

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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1answer
58 views

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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1answer
35 views

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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1answer
63 views

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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2answers
64 views

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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1answer
27 views

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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1answer
39 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
22 views

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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2answers
125 views

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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1answer
52 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
62 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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1answer
35 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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1answer
94 views

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
56 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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0answers
126 views

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
74 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
58 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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1answer
89 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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2answers
72 views

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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2answers
120 views

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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0answers
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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
254 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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2answers
237 views

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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0answers
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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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1answer
64 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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1answer
62 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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1answer
62 views

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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3answers
754 views

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 ...