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Questions tagged [conic-sections]

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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Distance of centre of an ellipse touching both the positive X and Y axes from origin when the ellipse is being rotated

If a horizontal ellipse touches the X-axis and Y-axis in the first quadrant, and this ellipse is rotated in anti-clockwise sense always touching the X-axis and Y-axis, till the ellipse becomes ...
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How to restrict only top range of ellipse function, and what is its domain?

I am trying to graph the function of an ellipse that is: $$1=\frac{x^2}{49}+\frac{(y+1)^2}{9}$$. I want to make the horizontal ellipse's range $y \leq 0.838$. So, when I also have to write the domain ...
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derivation of ellipse parameters

At various places on the Web (including Mathematics StackExchange) are various methods of calculating the semimajor and semiminor parameters of a ellipse $(a, b)$ from the location of a focus on the $...
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How to find locus for given condition?

I've been through all the thinking but i could not get to the answer. Please help me in this. See images for details
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Area of Intersection of Two Ellipses

Given two ellipses in space where: a & c are major diameters b & d are minor diameters h & k are x-axis centers j & i are y-axis centers r & s are the rotation of each ellipse ...
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Why do early math courses focus on the cross sections of a cone and not on other 3D objects?

Conic sections seem to get special attention in early math classes. My question is why do these cross sections of cones deserve more attention than those of, say, a rectangular prism, a cube, or some ...
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Need help with this word problem using hyperbolas and need the final answer in (x,y)

John was in the lead, Ed was 1.5 miles behind and Jeff was 2 miles behind John. Then they heard an explosion. John heard it first and Ed heard it a second later and Jeff heard it 1.5 seconds after ...
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multiple parabolas along the same line of symmetry with the same x-axis intercepts.

Given the equation of the parabola: $$ f(x) = (x-78)(x+218 ). $$ Is it possible to have more than one parabola with the same axis of symmetry: (-70), and the same x-axis intersects: ( 78 , -218 ) ?
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A chord is drawn from a point $P(1, t)$ to the parabola $y^2 = 4ax$ which cuts the parabola at A and B. If $PA.PB = 3|t|$, then find the range of t.

A chord is drawn from a point $P(1, t)$ to the parabola $y^2 = 4ax$ which cuts the parabola at A and B. If $PA.PB = 3|t|$, then find the range of t. I was able to solve the question using parametric ...
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Three chords drawn to an ellipse whose mid points lie on a parabola

Find the values of $\alpha$ for which three distinct chords from $(\alpha, 0)$ to the ellipse $x^2 + 2y^2=1$ are bisected by the parabola $y^2=4x$ Parametric form of the parabola is $(t^2,2t)$ ...
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What is the coordinate value after moving counterclockwise by distance $d$ from a coordinate on the ellipse?

Let me define an ellipse function as follows: Assuming $a \ge b$, $$ f(x,y) = \frac{(x-x_0)^2}{a^2} + \frac{(y-y_0)^2}{b^2} = 1,$$ where $(x_0,y_0)$ is the origin of the ellipse, and $a$ and $b$ are ...
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Do all functions represent a section of an n dimensional object by another object of n-1 dimension?

If $x^2 + y^2 = 4$ represents the section of a cone by a plane horizontal to the cone, a circle, and $y^2 - x^4 = 4$ represents the section of a cone by a plane vertical to the cone, a hyperbola, what ...
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How to check if a line is a tangent to a circle?

Is there a short and simple way to check if a line is a tangent to a circle, without complicated distance formulae? A solution to a question in my book says that for a circle $(x-at^2)(x-a/t^2) + y(y-...
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How would graphing a hyperbola work, when including the $v$ in the asymptote equation?

The equation of an asymptote can be either $$y=\pm\frac{b}{a}\sqrt{ (x-h)} + v.$$ The $v$ tends to be ignored as trivial, as the $x$ value tends to infinity, which implies that the approximate ...
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Is $a$ in $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ always the semi-major axis?

In the equation for an ellipse: $$ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 $$ is $a$ always the semi-major axis of the ellipse, even if it is less than $b$? My inference is that it isn't, but $a$ seems ...
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Parabola chord through (4a,0) subtends right angle at vertex?

The result of a question in my book hinges on the fact that for $y^2 = 4ax$ every chord passing through (4a,0) subtends a right angle at the vertex. It is suppoesed to be a standard result, but I have ...
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Find certain points on circle when viewing from different point

I have a circle like this: Where there is a known line passing through the centre of the circle. I want to find the points D and E on the image (left and right most part of the circle) when the ...
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How can I prove that $a^2 + b^2 = c^2$ in a hyperbola? [duplicate]

I have a question about hyperbolas as follows. Given this basic diagram of a hyperbola, with points $a, b, c$ (and their negatives respectively), how can I prove the fundamental formula $a^2 + b ^2 = ...
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Set up an integral for the circumference of an ellipse

I've been given a problem, which I'm unsure whether I'm answering it right, and wondered if someone can take a look and tell me if I'm on the right track and any guidance would be appreciated. My ...
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2answers
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Problem on tangent to the parabola.

Let PQ be a focal chord of the parabola $y^2= 4ax$. The tangents to the parabola at P and Q meet at a point lying on the line $y = 2x + a, a > 0.$ If chord PQ subtends an angle $\theta$ at the ...
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Tool/algorithmic library for determining equation of quadric surface through 9 given points

Is there any software that can calculate(and eventually plot) the quadric surface generated by 9 given points in 3D space? I know I can calculate that by defining 9 equations in Mathematica and ...
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How to find the equation of a hyperbola given the asymptote, equation of axis and a point

Given that a hyperbola has asymptote $y=0$, passes through the point $(1,1)$ and has axis $y=2x+2$, determine its equation. The answer arrived at is $\displaystyle{4xy+3y^2+4y-11=0}$. However, I have ...
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1answer
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Minimum value of a general two degree curve

Let F(x, y) = $ax^2 + by^2 + 2gx + 2fy + 2hxy + c$ To find its minimum value I use the following logic abd procedure : When you put a point (x, y) in equation of a conic it gives a positive value when ...
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2answers
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What is the probability that a randomly chosen point lies inside a given parabola?

A parabola (say $y^2=4ax$) divides the coordinate plane into two regions: one considered to be "inside" the parabola and the other "outside". However, both these regions are infinitely large. What is ...
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Area inside an irregular ellipse

I'm working through a geometrical problem where I need to find the area on the 2D plane between different values of $L$, where: $$L = \sqrt{(A^2 + x^2 + y^2)} + By\tag1$$ I'm struggling to find an ...
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3answers
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Eccentricity going to zero — Geometric definition conic

Given a straight line $D$, a point $F\notin D$ and a positive real number $e$, a conic is a subset of ${\cal P}_2$ defined as: $$ \mathcal{C}(e,F,D) = \{M\in {\cal P}_2,\, d(F,M)=e\,d(M,D) \} $$ where ...
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Number of rational points on ellipse [duplicate]

How to find the number of rational points on the circumference of ellipse $$ \frac{x^2}{9}+\frac{y^2}{4}=1 \,$$
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Asymptotes of the hyperbola $\frac{1}{2} + \frac{y}{5} + \lambda x(\frac{1}{2} + \frac{x}{10} + \lambda y) = 0$.

I am trying to find the asymptotes of the hyperbola $\frac{1}{2} + \frac{y}{5} + \lambda x(\frac{1}{2} + \frac{x}{10} + \lambda y) = 0$. I read through the answer here , but the equation of the ...
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1answer
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Minimum value of length of tangent of the ellipse $x^2/a^2 + y^2/b^2 = 1$, intercepted between the co-ordinate axes

I have taken a parameter $(a \cos c, b \sin c)$ where $c$ is the eccentric angle and the tangent passing through this point cuts the x-axis at the point $(a \cos c, 0)$ and y-axis at $(0,b \sin c)$. ...
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1answer
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Increasing Eigenvalue in $V_t = \lambda I + \sum_{s=1}^{t} A_s A_{s}^T$ as $t$ increases

I am reading a paper on bandits where it defines a matrix: $$V_t = \lambda I + \sum_{s=1}^{t} A_{s} A_{s}^{T} $$ where $\lambda$ is a scalar constant, $I \in R^{d \times d}$, $A_s \in R^{d}$ ...
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Three normal to the parabola $y^2=x$ are drawn through the point .

Three normal to the parabola $y^2=x$ are drawn through the point $(c,0)$ then $$\textrm {a}. c=\dfrac {1}{4}$$ $$\textrm {b}. c=1$$ $$\textrm {c}. c>\dfrac {1}{2}$$ $$\textrm {d}. c=\dfrac {1}{2}$$...
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Eccentricity of Pluto

With an eccentricity of $0.25$, pluto's orbit is the most eccentric in the solar system. The length of the minor axis of its orbit is approximately $1×10¹⁰$. Find the distance between Pluto and the ...
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Circle drawn on focal chord of a parabola

Is it possible for a circle with diameter as any focal chord of a parabola to cut the parabola at 4 points (2 being the extremities of the focal chord)? We were asked to find the product of the ...
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1answer
38 views

Which origin should be shifted to reduce given equation into one with linear term missing

Find the point to which origin should be shifted to reduce the equation $$3x^2 - 2xy + 4y^2 + 8x - 10y + 8 = 0$$ into one with linear term missing. So I just know that a linear term is where the ...
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Prove that the conic $x^2 - 4xy + y^2 -2x -20y -11 = 0$ is a hyperbola and find the centre $(h,k)$

I have to prove that the conic $$x^2 - 4xy + y^2 -2x -20y -11 = 0$$ is a hyperbola and find the centre $(h,k)$. I proved it is a hyperbola using discriminant $b^2-4ac $ and the answer was greater ...
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How to prove $f(z)=z+\frac{1}{z}$ maps circles with $r\ne 1$ onto ellipses?

How to prove $f(z)=z+\frac{1}{z}$ maps circles with $r> 1$ onto ellipses of the form $$\frac{x^2}{(r+\frac{1}{r})^2}+\frac{y^2}{(r-\frac{1}{r})^2}=1?$$ It is simple to understand the mapping to ...
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Oblate Spheroidal coordinate system graphic representation of ellipse

I am having difficulty understanding how to interpret the coordinate system proposed by Spencer. From his handbook Field Theory Handbooks ISBN 9783540184300 proposed a system with the domains ...
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Showing that the locus of point $N$ is $x^2+y^2=a^2$

Question: A point $P(a\cos\theta,b\sin\theta)$ lies on an ellipse with equation $$\varepsilon:\frac{x^2}{a^2}+\frac{y^2}{b^2}=1.$$The tangent to the ellipse $\varepsilon$ at point $P$ is perpendicular ...
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1answer
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What is the eccentricity of the hyperbola given by $3x^2+7xy+2y^2-11x-7y+10=0$?

What is the eccentricity of the hyperbola given by the following equation? $$3x^2+7xy+2y^2-11x-7y+10=0$$ what i try $$S = 3x^2+7xy+2y^2-11x-7y+10$$ $\displaystyle \frac{dS}{dx}=6x+7y-11=0$ and $\...
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1answer
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Hyperbola geometry proof

The points $(r\cos \theta, r \sin \theta)$ and $(s \cos(\theta + \frac{\pi}{2}), s \sin(\theta + \frac{\pi}{2}) ) $ lie on the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ with centre $O$. Show ...
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1answer
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Finding the equation of a parabola, given the length of a portion of a focal chord, and the angle the chord makes with the parabola's axis

Find the equation of the parabola on a picture if $|FL|=8$ units and $\angle KFO=60^o$. $F$ is given as the focus of the parabola. We know that this parabola passes through the point $(0,0)$, so if I ...
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1answer
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Getting equation of a curve which touches a given curve

There was a Q in my class, "From any point on $ b^4x+2a^2y^2=0$ pair of tangents PQ and PR are drawn to hyperbola $ x^2/a^2-y^2/b^2=1$. Prove that QR touches a fixed parabola and also find it's ...
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How might I quickly determine the equation of the parabola, given the coordinates of its focus and vertex?

I have this MCQ. Which of the following is the equation of the parabola with focus at $(1,2)$ and vertex at $(3,2)$ : A. $ y^2 - 4y + 8x - 20 = 0 $ B. $ y^2 + y + 8x -20 = 0 $ ...
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1answer
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Proof that any conjugate diameters of rectangular hyperbolas are reflections across an asymptote. (Hyperbolic Orthogonality)

I came across the Wikipedia page on conjugate diameters of ellipses, circles and hyperbolas, stating [T]wo diameters of a conic section are said to be conjugate if each chord parallel to one ...
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1answer
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Max possible area, of a rectangle shape where one side is a half circle. circumference of 100m

A picture of the shape! I recently took a maths test where one of the questions was just unsolvable for me. I'm going to try to make it as clear as possible, to not create confusion. The question ...
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0answers
28 views

Being given an hyperbola $H$ and a point $M$ on an asymptote, give a formula for the nearest distance from $M$ to $H$ being given parameters $a,b,c$

Let there is an asymptote line $$ y = \pm \frac{b}{a}x$$ and a hyperbola $$ \frac{x^2}{a^2}-\frac{y^2}{b^2}=1$$ what is the nearest distance from any point at the asymptote to hyperbola as a function ...
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1answer
56 views

How to find the farthest point on a ellipse from a point within an ellipse?

I was wondering if you could help me figure this out. I've been trying to write some code to calculate the farthest point on an ellipse $(150w, 85h)$ from a given point $(55x, 20y)$ within the ...
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3answers
676 views

Equivalence of different ways of geometrical multiplication

There are at least five ways to multiply two natural numbers $a$ and $b$ given as integer points $A$ and $B$ on the number line by geometrical means. Two of them include counting, the others are ...
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2answers
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Finding tangent to a circle with only one coordinate given

So I came across this MCQ: Which one of the following is the equation of a tangent to the circle $ x^2 + y^2 = 9: $ A. $ x = -1$ B. $ x = 4$ C. $ y = -4$ D. $ y =3$ E. ...
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1answer
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What is the difference between shifting functions and shifting parabolas?

For shifting a parabola to the right, do we write a "$+$" sign or "$-$" sign in the equation? Is this the same way for shifting a function, as well? Here is the equation of parabola just for the ...