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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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Find eccentricity of ellipse $9x^2 +4xy+6y^2-22x-16y+9=0$

I tried to find the eccentricity by factorising $$9x^2 +4xy+6y^2-22x-16y+9=0$$ in the form of perpendicular distance from two lines but it is getting too lenghty as the $xy$ term is involved. Is ...
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1answer
25 views

Mapping of an ellipse to an ellipse with different eccentricity that maps focal points to focal points

The title describes what I'm looking for: Is there a (canonical) way of mapping an ellipse (interior and boundary) to an ellipse with different eccentricity that maps focal points to focal points? (...
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2answers
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How to approach on this - finding minimum distance of point on the ellipse from the centre of it.

Question The minimum distance of any point on the ellipse $$x^2+3y^2+4xy=4$$ from its centre is ______. Attempt Converted the given expression into $$(x+2y)^2-y^2=4$$. But, this becomes equation of ...
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2answers
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Write the equation of the image of the parabola $y=x^2-2x+1$ under a translation $\vec{p}=\hat{i}+3\hat{j}$

Write the equation of the image of the parabola $y=x^2-2x+1$ under a translation $\vec{p}=\hat{i}+3\hat{j}$ $y=(x-1)^2$ is a parabola whose vertex is $(1,0)$ and focus is $(1,\frac{1}{4})$.I dont ...
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1answer
39 views

The range of values of $a$ such that…

Question The range of values of 'a' for which the common tangent to the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2}=1$ and the parabola $y^2=4x$ and their chord of contact can form an equilateral ...
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0answers
54 views

Prove the locus is a rectangular hyperbola.

Question Prove that the locus of the intersection of two equal circles which are described on two sides $EF$ and $EG$ of a triangle as chords is a rectangular hyperbola whose center is the midpoint ...
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Show that the locus of the middle points of chords of a parabola passing through the focus is a parabola. [on hold]

Show that the locus of the middle points of chords of a parabola passing through the focus is a parabola. I need this asap please, I have test tomorrow
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0answers
49 views

sections of regular functions $\mathcal{O}_X$ of the sphere $X = \{x^2 + y^2 + z^2 - w^2 = 0\}$ in projective space $\mathbb{P}^3$ [on hold]

I am trying to understand the sheaf of regular functions $\mathcal{O}_X$ in the case of the sphere. The machinery seems rather difficult to set up. Let't try using the following proposition from the ...
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How to project a circle onto an angled plane?

I have cut a hole in my ceiling to fit an 8 inch diameter circular duct perpendicular to my floor so it goes straight up. The ceiling is pitched at 15 degrees. I want to draw the shape that this ...
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2answers
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How do I find the length of the major and minor axes of the ellipse $\frac{x^2}{49}+\frac{y^2}{25}=1$?

How do I find the length of the major and minor axes of the equation $\frac{x^2}{49}+\frac{y^2}{25}$? For the vertices I got $(7,0)$ and $(-7,0)$ The foci: $(2\sqrt{6},0)$, $(-2\sqrt{6},0)$ ...
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1answer
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How do I convert an expression in terms of the general equation of a conic section to one in the equation of an ellipse?

In a major assignment I am to determine the semi-major axis of an elliptic orbit for the star S2 around Sagittarius A*. I found some data that I have used to fit the points to an ellipse - however the ...
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3answers
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Find the locus of perpendicular drawn from focus upon variable tangent to the parabola $(2x-y+1)^2=\frac{8}{\sqrt{5}}(x+2y+3)$

Find the locus of perpendicular drawn from focus upon variable tangent to the parabola $(2x-y+1)^2=\frac{8}{\sqrt{5}}(x+2y+3)$. My approach I am trying to convert above equation in parabolic form $\...
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1answer
21 views

Where does a chord of an Ellipse equal to the length of the minor axis but running parallel to the major axis cross the minor axis.

Given an Ellipse, I need to know where a chord equal to the length of the minor axis but running parallel to the major axis cross the minor axis.
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How do I find the vertices, foci, eccentricity, and the lengths of the minor and major axes of the following ellipse? [on hold]

How do I find the vertices, foci, and eccentricity of the following ellipse? $$\frac{x^2}{49} + \frac{y^2}{25}=1$$ I put the ellipse in the standard equation so far, but am not sure what to do in ...
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1answer
52 views

Solving for Ellipse Parameters Given a radius and angle (Challenge 2)

Given an ellipse centered on the origin in an x-y plane expressed as $$\bigg(\frac{x}{a} \bigg)^2+\bigg(\frac{y}{b} \bigg)^2 = 1$$ In polar coordinates with radius $R$ and angle = $\theta$, this can ...
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1answer
39 views

Solving for Ellipse Parameters given a radius and angle (Challenge 1)

Given an ellipse centered on the origin in an x-y plane expressed as $$\bigg(\frac{x}{a} \bigg)^2+\bigg(\frac{y}{b} \bigg)^2 = 1$$ In polar coordinates with radius $R$ and angle = $\theta$, this can ...
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0answers
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computes the $\phi$ angle of an ellipse

I've an ellipse that is rotated and not centered in the origin. Hence, I've an ellipse equation that is in the form of: \begin{equation} \frac{((x-h)cos(\phi) + (y-k)sin(\phi))^2}{a^2} + \frac{((x-h)...
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How to determine what kind of conic section in the affine plane?

So, I've been struggling a bit with understanding this problem. Let $P^2$ be the real projective plane with homogenous coordinates $(x_0:x_1:x_2)$ Let $\cal{C}$ be the line given by $$x_0^2 + 2x_0x_1+...
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2answers
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Points closest to the edge of a square

Let S be the square formed by the four vertices (1,1),(1,-1),(-1,1), and (-1,-1). Let the region R be the set of points inside S which are closer to the centre than to any of the four sides. Find the ...
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1answer
36 views

Determine the polarities of a self polar triangle

Consider a triangle $PQR$, $P(0,2,1), Q(1,0,2), R(0,4,9)$. Determined the polarities if triangle $PQR$ is self polar. By definition of self polar triangle, point $P$ gets mapped to line $QR$, point $...
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Prove $AX$, $BY$, $CZ$ concur

$\Delta ABC$. Let $P,Q$ be two points lies in the interior of the triangle. Let $AP,BP,CP$ intersect $BC,CA,AB$ at $X_P,Y_P,Z_P$. Point $X_Q,Y_Q,Z_Q$ are defined similarly. Let $T_P$ be a point lies ...
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1answer
26 views

Prove that the parabloas are mutually perpendicular.

Given that two parabolas have the same focus with their axes of symmetry in opposite directions. Then I have to prove that the two intersect at right angles. As I think, because the axes are ...
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0answers
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Angle of the slope of an ellipse

Find the relation between the angle made by a straight line connecting the origin and an ellipse (angle made between that line and the x axis) and the slope of the tangent to the ellipse. This should ...
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1answer
28 views

Coordinate Geometry - Parabola [closed]

Problem: A quadrilateral is inscribed in a parabola, then (A) Quadrilateral may be cyclic. (B) Diagonals of the quadrilateral may be equal. (C) All possible pairs of adjacent side may be ...
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1answer
41 views

Finding Eccentricity of A Hyperbola

Given an asymptote to an hyperbola and that a line perpendicular to it, intersects it at a single point, we need to find its eccentricity. Asymptote : $5x-4y+5=0$ and Tangent : $4x+5y-7=0$. I ...
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1answer
16 views

If the parabola is translated from its initial position to a new position by moving its vertex along the line $y=x+4,$

The parabola $y=4-x^2$ has vertex $P.$It intersects $x-$axis at $A$ and $B.$ If the parabola is translated from its initial position to a new position by moving its vertex along the line $y=x+4,$ so ...
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1answer
37 views

Drawing bezier curve from a parabola

I'm not a math guy, sorry. I read posts on the subject but couldn't find the answer to my problem (or didn't understood the answers). I'd like to get a simple answer. I know a generic parabola ...
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2answers
36 views

What is $b$ in this “conic general form” equation of a circle? $x^2+y^2+4x-4y-17=0$

Take the equation of a circle from this khanacademy video as an example: $$x^2+y^2+4x-4y-17=0$$ $$a=x^2$$ $$b= ? $$ $$c=y^2$$ $$d=4x$$ $$e=-4y$$ $$f=-17$$ What is b equal to, and why did we "jump" ...
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3answers
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Is there a parabola which is similar to a branch of hyperbola?

Parabola and a branch of hyperbola, visually looks similar. The only difference I find is that, when x tends to infinity, hyperbola approaches a straight line (asymptote). Whereas if I draw an ...
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1answer
28 views

Minimum distance of curve from origin

I have a parabola $(y+5)^2 = 4x$ and I need to find its minimum distance from origin. Scientific calculators aren't allowed. I have tried : 1) Substituting parametric coordinates $(r\cos Q, r\sin Q)$...
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1answer
38 views

Divergence Theorem - Cone

Here's the question: Evaluate the surface integral $\iint _S F\cdot n \space dA$ by the divergence theorem. $ \mathit F = [xy, yz, zx]$, S the surface of the cone $x^2 + y^2 \le 4z^2, \space \space 0 ...
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2answers
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Inside and outside of an ellipsoid

Let $A$ be a positive definite matrix. The equation $x^TAx=1$ defines an ellipsoid. I would like to justify the fact that $x^T A x > 1$ implies that $x$ is outside of the ellispoid. In other ...
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3answers
553 views

Area of a triangle inside an ellipse

$F_1$, $F_2$ are are foci of the ellipse $\dfrac{x^2}{9}+\dfrac{y^2}{4}=1$. $P$ is a point on the ellipse such that $|PF_1|:|PF_2|=2:1\;$, then how could I figure out of the area of $∆PF_1F_2$? As ...
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2answers
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Fourth point of intersection of two conics

Five points in general position define a unique conic section. Let $Q_1$ be a conic through points $A,B,C,E_1,F_1$ and likewise $Q_2$ through $A,B,C,E_2,F_2$. Two conics (over an algebraically closed ...
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0answers
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Approximate a ($y²-x²=1$) hyperbola with line segments and elliptic ($a(x-x_0)²+b(y-y_0)²=1$) arcs

On some IT graphic systems, you have tools which draw line segments and circle or elipse arcs, but which do not draw parabolas or hyperbolas, and in many cases, those system keep track of graphical ...
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3answers
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How to calculate the tangent of a 3d Parabola

I have the following parabola $$ P:y^2 − 6x − 6y + 3 = 0.$$ How can I find the tangent parallel to line $\ell: 3x − 2y + 7 = 0$? I wouldn't have any problem with this problem if there was only one ...
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2answers
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Finding equation of an ellipse given four points [closed]

I am just wondering how can I find the equation of an ellipse given that it passes through these four points? A(1,1) B(3,4) C(1,7) D(-1,4) Appreciate any solutions and answers.
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1answer
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Transform parabola to be tangent to line in point and through other point

Sorry for stupid question, but I give up. I've spent whole weekend to solve that and no results. Please help. I think I know the solution, but it doesn't work form me. So now I am not sure. I am not ...
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0answers
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Find the area bounded by the two Ellipses $E_1$ and $E_2$

$E_1$ is an ellipse with center at $(0,0)$ and major axis alone the line $3x+y=0$ with eccentricity $e_1=\frac{\sqrt{3}}{2}$ $E_2$ is another ellipse with center at $(-2.5,10)$ and Major axis alone ...
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1answer
34 views

Level curves of $f(r) = \sum_{i=1}^na_id(r,r_i)$ with $r_i\in \mathbb{R}^n$

Let $$f(r) = \sum_{i=1}^na_id(r,r_i)$$ Where each $r_i \in \mathbb{R}^n$ and each $a_i\in \mathbb{R}$ and $d:\mathbb{R}^n\times\mathbb{R}^n\rightarrow\mathbb{R}$ denotes the usual distance function. ...
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Why is the semicubical parabola considered a parabola?

Simple question here. Why is the semicubical parabola ---defined by the equation $y^2 = a^2 x^3$--- considered a parabola? If you look at the graph, it looks like a half parabola at best. This ...
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2answers
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Ellipse on a Circular Cylinder in Cylindrical Coordinates

Is there an equation in cylindrical coordinates for an ellipse (tilted at some angle) on the surface of a right circular cylinder of radius r? For simplicity, I envision the cylinder to be coincident ...
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The equation of the tangent to conic $x^2 - y^2 - 8x + 2y + 11 = 0$ at $(2,1)$ is?

Options are :- A) $x+2=0$ B) $2x+1=0$ C) $x-2=0$ D) $x+y+1=0$ I have tried using differentiation , find out the slope But, its 0. So, I'm not getting my answer correct.!!?
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1answer
25 views

Can quadric surfaces be made by cutting a 4-dimensional cone?

In my high school multivariable calculus class, we recently learned of quadric surfaces. Since they appeared to be a generalization of conic sections to 3 dimensions, I wondered if they could be ...
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2answers
41 views

How to make mirrored parabola

I try to figure it out how to make mirror of the parabola to some line. For example like that: In that example my original parabola is: $f_p(x) = x^2\ \ $ - (red line) My mirror center line is: $...
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1answer
72 views

parabola equation from two points and vertex

For a optimisation task I'm facing a problem that my limited geometrical knowledge can't solve. I have a cloud of points in a 2d plane, where I want to perform some binary metric on to select a ...
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1answer
18 views

Find the ratio of the area of the region bounded by the parabola and the line segment $PQ$ to the area of triangle $PQR$

For the parabola $y=-x^2$,let $a<0,b>0,$ $P(a,-a^2),Q(b,-b^2)$.Let $M$ be the mid point of $PQ$ and $R$ be the point of intersection of the vertical line through $M,$ with the parabola.Find the ...
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3answers
41 views

Points $A$ and $B$ lie on the parabola $y=2x^2+4x-2,$such that the origin is the mid point of the line segment $AB$

Points $A$ and $B$ lie on the parabola $y=2x^2+4x-2,$such that the origin is the mid point of the line segment $AB$.Find the length of the line segment $AB$ $y=2(x^2+2x-1)=2(x+1)^2-4\implies (y+4)=2(...
4
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1answer
23 views

Are these answers regarding focus and directrix correct?

1. Find the vertex, focus and directrix for the parabola given by $y = 2x^2$ $(y-0) = 2(x-0)^2$ $(x-0)^2 = \frac{1}{2}(y-0)$ $4p = \frac{1}{2} \implies p = \frac{1}{8}$ V:$(0,0)$ F: $(0,\frac{1}{...
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2answers
35 views

Locus of Midpoints of chords in a circle.

This question is a Conics/Locus problem: The circle $x^2+y^2=25$ cuts the y axis above the x axis at A. Find the locus of the midpoints of all chords of this circle that have A as one endpoint. I’ve ...