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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9
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1answer
273 views

Right triangle on an ellipse, find the area

Beginning note: Please wait until the animations load. The loading might take some time depending on your internet connection. Secondly, the title and the content of the question might not be well ...
0
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2answers
15 views

How to find the solutions for the quadratic equation for conic sections $\epsilon \in (0,1)$

Going from this definition of the conic section: $\epsilon |Pl| =|PB|$, you get the following equation for the intersection with the $x$-axis: $y^2 = (\epsilon ^2-1)x^2+(B-\epsilon ^2L)2x+\epsilon ^2L^...
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0answers
9 views

Find a point on an ellipse given a different point on the ellipse and either an arc length or a chord length.

Given a point $A$ on an ellipse and an arc length, how can I find the resultant point $B$ on the ellipse such that point $B$ is the arc length away from Point $A$? Alternatively, given a point $A$ on ...
0
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0answers
47 views

Double Integral over the region of an ellipse cut off by a circle

I've been stuck on this question for awhile. I need to calculate the double integral $\iint_R \frac{1}{r^3} dA$ using polar coordinates. R is the region displayed below: The ellipse has centre (4,...
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0answers
16 views

Ellipsoid Axis at density contour, why choose biggest eigen value for axis?

I've been trying to figure out how to find the density contour for a multivariate normal density function with an arbitrary number of dimensions. I've found a lot of examples for 3Dimensions and for ...
0
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1answer
26 views

Find cone-plane intersection points in a construction

I have two points on the X axis, A and B, which are connected to the two points, C and D on the sketch plane parallel to XY plane. I have a point E which lies at distance h from D point in Y direction ...
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2answers
42 views

Prove that the directrix is tangent to the circles that are drawn on a focal chord of a parabola as diameter.

Question: Prove that the directrix is tangent to the circles that are drawn on a focal chord of a parabola as diameter. Here is a picture; What I have attempted; Let the parabola ...
1
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0answers
35 views

Find the equation of a hyperbola, given a point on it and the length of the transverse axis

My textbook has the following question: The transverse axis of a hyperbola is of length $24$ and the curve passes through the point $(13, 10)$. Find the equation of the hyperbola. Also give the ...
0
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2answers
71 views

How to get the properties of an ellipse with six points given.

I am looking for a way to calculate the lengths of both semi-axes and the rotation angle of the ellipse in the image as shown in this picture. Six points are given, with two pairs of points being ...
0
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0answers
18 views

Point halfway around ellipse quadrant

I want to find the length between the centre of an ellipse and a point, P, on the ellipse, where the arc length between P and the intersection of the semi-minor axis with the ellipse is equal to the ...
0
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0answers
14 views

Derive the parametric form of the locus of point where difference between distance to two points is constant

Given two points $P_1=(x_1,y_1)$ and $P_2 = (x_2,y_2)$, the locus of the point whose (signed) difference between the distance to the two points is a constant $\Delta$ is one branch of a hyperbola ...
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0answers
27 views

Hyperbola equation proof

I've been trying to prove the canonical form of the hyperbola by myself. $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 $ I started from the statement that ...
0
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1answer
37 views

How to derive formula for focus of a parabola?

I understand how to obtain the formula for the vertex of a formula, $ y= a(x-h) + k $ where $ h=-b/2a$ and the vertex is $(h,k)$. However I don't know how to get to $(h,k+1/4a)$. Could someone please ...
0
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2answers
57 views

Ellipse - relation between a and b such that $F_1P \perp F_2P$

Consider the ellipse $\displaystyle \dfrac{x^2}{a^2} + \dfrac{y^2}{b^2}=1$ with foci $F_1 (-e, 0)$ and $F_2 (e, 0)$ (where $e$ is the linear eccentricity). What is the relation between $a$ and $b$ so ...
0
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2answers
40 views

How to determine standard equation of a conic from the general second degree equation?

From a given general equation of second degree i can determine the conic by following rules: Given equation: $ax^2+by^2+2hxy+2gx+2fy+c=0$ then if, $abc+2fgh-af^2-bg^2-ch^2$ is not equal to zero ...
0
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0answers
58 views

Computing the properties of the 3D-projection of an ellipse.

I have an ellipse that is rotated around the white axis (see image below) in 3-dimensional space by an angle α. The axis passes through the perimeter and one of the ...
0
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1answer
20 views

Ellipse from two arbitrary points, tangent at P1 and a focal point

Is it possible to find this? Really only need the semi major axis or even it's orientation. Please see the linked image. Known elements are in red and the desired element is in blue.
2
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2answers
44 views

Finding the tangent of an ellipse that is perpendicular to a line

The books say's "Find the equations of the tangents to $x^2+3y^2=4$ which are perpendicular to the line $x-2y=7$" I've graphed them and found that the given line does not pass through the ...
0
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1answer
42 views

circle cuts three circles at the extremities of the diameter

If the circle $$x^2 + y^2 + 2gx + 2fy + c = 0$$ cuts the three circles $$x^2 + y^2 – 5 = 0\space;\space x^2 + y^2 – 8x – 6y + 10 = 0 \space;\space x^2 + y^2 – 4x + 2y – 2 = 0;$$ at the ...
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3answers
35 views

Algorithm: Intersection of two conics

I am looking for a detailed description of an algorithm for the classical problem of computing the intersection of two conic curves. The curves are given by two equations of the form: $$ a x^2 + b y^...
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0answers
20 views

$(x-1)(y-2)=5$ and $(x-1)^2+(y+2)^2=r^2$ intersect at four points $A,B,C,D$. Centroid of $\Delta ABC$ lies on $y=3x-4$, then the locus of $D$

$(x-1)(y-2)=5$ and $(x-1)^2+(y+2)^2=r^2$ intersect at four points $A,B,C,D$. If centroid of $\Delta ABC$ lies on $y=3x-4$, then what is the locus of $D$? I did try a couple of things, but I honestly ...
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0answers
37 views

Volume of paraboloid that is cut with plane

How to calculate the volume of the paraboloid : x^2 + y^2 = z that is cut with x + y + z = 5 plane. Please give several methods if you can. Thank you very much for answers.
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1answer
24 views

Condition for common tangents to Circle and parabola

The parabola $y^2=4ax$ and circle $x^2+y^2+2bx=0$ have more then one common tangents;, Then which one is/are right, $(a)\; ab>0\;\;\;\; (b)\; ab<0\;\;\;\; (c)\; ab<-2\;\;\;\; (d)\; ...
0
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1answer
15 views

Equation of tangent at vertex

The equation of tangent at vertex of parabola $4y^2+6x=8y+7$ is .. I simplified the equation and got $4(y-1)^2 =-(6x-11)$. What do I do further?
1
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1answer
22 views

Focus of a parabola

If (2,0) is the vertex and y-axis the directrix of a parabola find the focus of the parabola. What does y-axis is directrix mean here?
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2answers
29 views

Partial differentiation and tangency

$\text{Q}$ Write the equation of tangent at the vertex of the parabola $2y^2+3y+4x-3=0$ . $\text{My attempt}$ if I partially differentiate the curve with y then I get $y+0.75=0$ which is correct ...
7
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2answers
49 views

Show that the curve $\dfrac{x^2}{a^2} +\dfrac{ y^2}{b^2} = 1$ form an ellipse

If the definition of an ellipse is the set of points $(x,y)$ such that given two focus points $F_1, F_2$ the sum of the distances from $(x,y)$ to each focus point is constant, how can one show that ...
0
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1answer
35 views

How to proove that foot of perpendicular drawn from focus to any tangent of an ellipse lie on auxillary circle?

One way is to find the foot of perpendicular and directly putting it into the equation of auxiliary circle. But that is quite a lengthy proof, is there any other short method to prove this property?
2
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1answer
69 views

What practical purpose — or application — do directrices serve?

In Calculus II (and briefly in Trigonometry, if I remember correctly) the concept of a directrix began poking its head around conic sections. While covering parabolas, ellipses, and hyperbolas, the ...
2
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2answers
38 views

Can a line be a taxicab parabola?

For example, a line segment can be a taxicab ellipse if the sum of the distances equals the distance between the foci. So, can a line be a taxicab parabola?
6
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2answers
114 views

Locus of a point on a fixed-length segment whose endpoints slide along orthogonal lines

Suppose we have some segment $AB$ of constant length that slides in such a way that its endpoints are moving along orthogonal lines. Let $P$ be a point in the segment so that $|AP| = a$ and $|PB| = b$....
0
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1answer
29 views

Can a line parallel to axis of parabola also represent tangent at a point along with the one whose slope is found using calculus?

Consider a parabola with the equation $y^2=4x$ its axis is the x-axis and vertex is (0,0) and focus at (1,0). Consider any point on the parabola say (4,4). Now we define tangent at this point as a ...
0
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0answers
38 views

HYPERBOLA : Problem [duplicate]

If two points $P$ and $Q$ on the hyperbola $\frac{x^2}{a^2} -\frac{y^2}{b^2} = 1$ whose centre is $C(0,0)$ are such that $CP$ is perpendicular to $CQ$ , $a<b$ , then prove that $$\frac{1}{(CP)^2} ...
0
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1answer
20 views

equation of an ellipse given its center and two tangent lines

There exists an ellipse centered at (0,0) with two tangent lines given by $y=-\frac{1}{2}x + \frac{\sqrt{39}}{2}$ and $y=\frac{1}{3}x + \frac{7}{3}$. Find the ellipse. So, I used the equation of one ...
0
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1answer
33 views

Eccentricity of a hyperbola given the angle between the x-axis and its asymptote

I need to find the eccentricity of a hyperbola whose asymptote makes an angle $\alpha$ with the $x$-axis. So, I take the case where the transverse axis will be horizontal, $i.e.$ $\frac{x^{2}}{a^{2}}...
0
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1answer
40 views

Ellipse and chord length

There is a analytic geometry problem: In the ellipse $\frac{x^2}{4}+y^2=1$, segment $AB$ is a chord and $AB=\sqrt{3}$, find the maximum and minimum area of $\triangle AOB$. My progress Assume ...
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0answers
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Program for General Form

I am trying to find the general form of the conic sections, however, I am not able to create a program that runs it. I know the way you find it is by using B^2-4AC but I would like to write a program ...
13
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8answers
423 views

How do I transpose an ellipse function to stretch the ellipse into curved space?

I'm working on an engineering project, using CAD software. I can write simple parametric functions to draw an ellipse, with $\theta$ ranging from $0$ to $2\pi$ radians:$$x=3.75\cos\theta$$$$y=1.25\...
0
votes
1answer
57 views

Ellipse's farthest point to another point

I am trying to find the farthest and closest points of a ellipse without using any brute force type of coding. The processing power is limited so it should be as pinpoint as possible. I have tried a <...
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0answers
10 views

Calculating X & Y coordinates of a point that is perpendicular to an ellipse point AND offset by -5

I am trying to calculate an offset point from a point on an ellipse - I need to be perpendicular to each point on the ellipse but 5 points in from the point on the ellipse. The result will probably ...
5
votes
4answers
230 views

Find the latus rectum of the Parabola

Let $y=3x-8$ be the equation of tangent at the point $(7,13)$ lying on a parabola, whose focus is at $(-1,-1)$. Evaluate the length of the latus rectum of the parabola. I got this question in my ...
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0answers
28 views

Graphing calculator leaving gaps in a drawn graph of a rotated parabola

I am graphing the equation of two rotated parabola on the graphing calculator and, after finding the y= form for each using the quadratic formula and entering them into a program to graph them they ...
1
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3answers
35 views

Show that the tangent to the hyperbola $(x_0 , y_0)$ does not intersect the curve anywhere else.

Question: Consider the hyperbola $ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ Given that the equation of the tangent at the point $(x_0 , y_0)$ is $\frac{xx_0}{a^2} - \frac{yy_0}{b^2} = 1$ ...
0
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4answers
70 views

What region in $\mathbb{C}$ does $\left|{z-1}\right|+\left|{z+1}\right|$ = 2 describe?

I have played around with this a bit and keep getting something that doesn't seem right. Perhaps I'm overlooking something. Using the definition of distance in the complex plane I transform my ...
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2answers
75 views

What is condition for second degree equation to represent a pair of straight lines?

According to my text the necessary and sufficient condition for a general equation of second degree i.e. $ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0$ to represent a pair of straight lines is that 1) the ...
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1answer
39 views

How to calculate an inner ellipse points that is always a set distance from an outer ellipse points

I have an Ellipse with known coordinates , I would like to know how I can create an inner ellipse coordinates that are exactly 5 inches perpendicular from the outer ellipse points. Please see the ...
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3answers
31 views

Why does this limit of hyperbolic cosines equate to a parabola?

I bolded my main question below, and I would like to understand why the following limit is true: $$\lim _{ n\rightarrow { 0 }^{ + } }{ \frac { \cosh { (nx) } -1 }{ \cosh { n } -1 } } = { x }^{ 2 }$$ ...
0
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0answers
14 views

Plot the path of two axis relating to points on a super ellipse?

I have two axis linked to each other; and I need to move both axis 1 and axis 2 to follow the path of a known super ellipse - I have 38 points defined for the super ellipse. A picture is attached, I ...
1
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1answer
74 views

Find the second focus of an ellipse given one focus, the point furthest from it, an arbitrary other point on the ellipse [duplicate]

I have the following three points: A: One focus of an ellipse C: The furthest point from this focus (the far side of the major axis) D: Some other arbitrary point on the ellipse From there, it's ...
0
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1answer
36 views

Map an ellipsoid to a sphere

If I have a ellipsoid described by: $(\boldsymbol{x} - c)^T \boldsymbol{A} (\boldsymbol{x} - c) = 1$ How do I get the transformation to an unit sphere centered at the origin? From the principal ...