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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Compute the shortest possible distance between a point on the graph of $f$ and a point on the graph of $f^{-1}$

Let $f(x)=(x+3)^2+\cfrac{9}{4}$ for $x\ge -3 $.Compute the shortest possible distance between a point on the graph of $f$ and a point on the graph of $f^{-1}$. My effort Let $P,Q$ be points on the ...
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Imaginary tangents of parabola

For a parabola $y^2 = 4ax$ ,we can draw $2$ tangents from any point.If the point is outside of parabola then obviously we can draw $2$ tangents. If the point is on the parabola then the two tangents ...
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What simple topological properties of conic sections can be explored?

In the framework of my science fair project I am working on conic sections in different metric spaces. What simple topological properties/operations and so can I explore on them? Edit: To clarify, ...
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How to determine the reflection point on an ellipse

Here is my problem. There are two points P and Q outside an ellipse, where the coordinates of the P and Q are known. The shape of the ellipse is also known. A ray comming from point A is reflected by ...
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Co-ordinate Parabola Circle Contained in it; Difference in maximum and minimum possible radius

If the Difference of radii of larget and smallest Circle passing through the focus of Parabola $$Y^2=4x$$ and toughing parabola in at least one point is My Approach Let Circle be $$C: ...
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Ellipse set with one fixed focus, co-tangential at origin

Find equation of an ellipse tangential to x-axis at origin and whose one focus is fixed at P $ (-a,-b), $ another is variably placed at Q $ (a\, m, b \,m).$
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An equation of a tangent to the parabola $y^2=8x$ is $y=x+2$. the point on this line from which the other tangent

Problem : An equation of a tangent to the parabola $y^2=8x$ is $y=x+2$. the point on this line from which the other tangent to the parabola is perpendicular to the given tangent is given by ... ...
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Determining the angle degree of an arc in ellipse?

Is it possible to determine the angle in degree of an arc in ellipse by knowing the arc length, ellipse semi-major and semi-minor axis ? If I have an arc length at the first quarter of an ellipse and ...
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GetThere Airlines currently charges $200$ dollars per ticket.How can they maximixe their revenue if they were to increase the price?

GetThere Airlines currently charges $200$ dollars per ticket,and sells $40,000$ tickets.For every $10$ dollars they increase the ticket price,they sell $1000$ fewer tickets. How much ...
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364 views

Is it possible to calculate the volume of a parabolic arch?

Given that you know the equation of a parabola that only has positive values, is it possible to find the volume of the parabolic arch itself? NOT the volume of space underneath the arch. I asking ...
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Partially differentiating the equation of a conic section

There was this question where a double degree equation of a conic section was given and the coordinates of the center of conic had to be found. The solution first partially differentiated the equation ...
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1answer
270 views

Finding volume of enclosed region

The base of S is the region enclosed by the parabola $y = 9 − 9x^2$ and the X - axis. Cross-sections perpendicular to the X - axis are isosceles triangles with ...
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1answer
58 views

Find a parabola knowing its distance from a point.

I have the parametric parabola: $$ y=f(x)=C(x-4)(x-5)+D $$ where $D$ is fixed. I want to find for which value of $C$ the distance from the parabola to the point $(4,0)$ is exactly $\frac{1}{3}$ and ...
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Finding the eccentricity/focus/directrix of ellipses and hyperbolas under some rotation

Suppose I have an ellipse/hyperbola rotated about the origin by some angle $\theta$. Am I right in saying that the following general process will find the eccentricity $e$ of these conics? Find ...
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Determining the major/minor axes of an ellipse from general form

I'm implementing a system that uses a least squares algorithm to fit an ellipse to a set of data points. I've successfully managed to obtain approximate locations for the centre of the ellipse but I ...
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1answer
28 views

No. of points determining a unique parabola

For a parabola, let Focus: $(a_1,b_1)$ Equation of directrix: $y-mx-c=0$ The equation of parabola is, $\sqrt{(x-a_1)^2+(y-b_1)^2}= \frac{|y-mx-c|}{\sqrt{1+m^2}}$ There are 4 parameters ...
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“Mean” ellipse inbetween two ellipses

I am dealing with two ellipses, described by bigger one: 30052069549920 - 560534420160 x + 3754285920 x^2 - 84631979520 y + 18247680 x y + 177708960 y^2 == 0 smaller one: -1431356032960 + ...
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Finding x-intercept of a parabola given one x-intercept

I am given an $x$-intercept of $-3-\sqrt{7}$ and I am asked to find the other intercept. I am having trouble since I don't have any other information but the given $x$-intercept. My guess is that the ...
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A circular paraboloid can be a elliptic paraboloid?

I'm aware of this similiar question: what is the difference between an elliptical and circular paraboloid? (3D) But I need help in a different way. In my calculus exam, I was asked to name the ...
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high of a parabola bridge [closed]

My problem is: A bridge has the shape of a parabola with the equation $$x^2=-48y$$ and its arc length is $$l=24m$$ How to compute the heigh of the bridge without using integrals? I mean with an ...
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Parametrization of $K$-rational points of the hyperbola

Let $K$ be a perfect field of characteristic $\neq 2$ and consider its algebraic closure $\overline K$. Moreover define $$C=\{(x,y)\in \mathbb A^2(\overline K)\,:X^2-Y^2=1\}.$$ How can I get the ...
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467 views

Equation of the locus of centre of the ellipse?

An ellipse slides between two perpendicular lines. To which family does the locus of the centre of the ellipse belong to?
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How to find equation of parabola when we only know the equation of latus rectum and coordinates of vertex?

Suppose the equation of latus rectum is x=4 and the vertex is (2,3). I am confused wouldn't there be many parabola with this same vertex and latus rectum.If not how to find the equation? The answer ...
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How to find the eccentricity of this conic?

How to find the eccentricity of this conic? $$4(2y-x-3)^2-9(2x+y-1)^2=80$$ My approach : I rearranged the terms and by comparing it with general equation of 2nd degree, I found that its a ...
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64 views

Show that an infinite number of triangles can be inscribed in either of the parabolas $y^2=4ax$ and $x^2=4by$ whose sides touch the other parabola.

Show that an infinite number of triangles can be inscribed in either of the parabolas $y^2=4ax$ and $x^2=4by$ whose sides touch the other parabola. I tried to solve it but failed.Can someone please ...
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1answer
27 views

Vertex Form of Parabola - Why does it work?

Recently, I have been trying to plot parabolas of quadratic equations. First, I have to convert them to vertex form and then we can easily plot them. This makes me wonder why the vertex form of a ...
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372 views

maximum radius of a circle inscribed in an ellipse

Consider an ellipse with major and minor axes of length 10 and 8 resp. The radius of the largest circle that can be inscribed in this ellipse, given that the centre of this circle is one of the focus ...
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1answer
53 views

Representing transformed ellipse

I am drawing ellipses using SVGs. An ellipse is described by center {x,y}, radiusX and radiusY. To be able to draw every ellipse, I also added rotate angle alpha. (As described here - every ellipse ...
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1answer
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How to compute the chord length of an ellipse?

How do I calculate the chord length of ellipse? I need to design a blade vane rotating inside an elliptical profile. The blade is supposed to be in contact with ellipse with a maximum clearance of ...
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1answer
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Tracing of a conic

I have my assignment of drawing a parabola with equation $y^2=16x$ . I cannot see how to do it. I cannot see any parameter to draw a parabola . One of my friends said use latus rectum but as I am a ...
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Sketching a parametrised cone and a geodesic lying on it.

I just started a new module at University and I am having some trouble with parametrisation. I am given a parametrisation of a geodesic lying on a cone in notation $r(t)=x(t){\bf i}+y(t){\bf ...
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Conics (Ellipse): Complete the Equation to Give at least 1 point

The question asks: For which values of $a$ does the conic $4x^2+16x+5y^2-40y=a$ have at least one point? (State your answer in interval notation.) $a\in$ ___ I was able to understand that ...
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Help with Conic: Hyperbola's chord of contact

please help with this proof. "Show that the tangents at the endpoints of a focal chord of the hyperbola $ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 $ meet on the corresponding directrix." This is a ...
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If an parabola has its focus at the (a,b) and has directrix at x=c…

If an parabola has its focus at the (a,b) and has directrix at x=c, what would the equation 4p(x – h) = (y – k)^2 look like in terms of a,b, and c?
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Eccentricity of a general ellipse

How to find the eccentricity of an ellipse $5x^2 + 5y^2 + 6xy = 8$ ?. I tried it by factorizing it into the distance form for a line and point but I failed. Please help
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Foot of perpendicular on a chord of a conic

For a standard ellipse, a chord subtends an angle of $90^{\circ}$ with the centre $(0,0)$ . To find the locus of the foot of perpendicular to this chord from the centre of the ellipse, I wrote the ...
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What is the equation of the bottom half of the parabola $x + (y - 2)^2 = 0$?

A parabola has the equation: $$x + (y - 2)^2 = 0$$ I can't find the $y$ without getting the equation into some weird recursion.
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Parametrize an intersection of a plane and an elliptic paraboloid

I'm supposed to parametrize the intersection of the plane that has the equation $z = 5x + 3y$ and the 'elliptic paraboloid' with the equation $z = 3x^2+2xy+3y^2$ These two equations can also be ...
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Distances in HP

A variable straight line passes through the fixed point $A(6,1)$ and meets the ellipse $x^2 + 2y^2 = 2$ at points $B$ and $C$. If $P$ is a point such that the lenghts $AB, AP, AC$ are in HP (harmonic ...
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Point of intersection of tangents

If the distance of two points $P$ and $Q$ from the focus of of a parabola $y^2 =4ax$ are $4$ & $9$ then what is the distance of the point of intersection of tangents at $P$ and $Q$ from the ...
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Number of inscribed triangles in a rectangular hyperbola touching a parabola

How many triangles can be incribed in the rectangular hyperbola $xy= c^2$ whose sides all touch the parabola $y^2 =4ax$. How can we start the question . Please help.
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Locus of vertex

A variable parabola of latus rectum $l $, touches a fixed equal parabola , the axes of the two curves being parallel . Then locus of the vertex of the moving curve is a parabola , then what is the ...
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How to change $ Cx^2 + Dy^2 + Ex + Fy + G = 0$ to$ (x-h)^2/a^2 ± (y-k)^2/b^2=1 $ using only the variables C, D, E, F, and G

Or, state the terms a,b,h,and k in terms of C, D, E, F, and/or G $Cx^2 + Dy^2 + Ex + Fy + G = 0$ $(x-h)^2/a^2 ± (y-k)^2/b^2=1$
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Entire function assume real values for $z=x^2+ix$

Let $f$ be an entire function such that $f(z)$ is real for $z=x^2+ix$. Is there exists such a function which is not constant? Previously I thought that ...
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Axes Rotation Problem

Given $$x^2 - 4xy + 5(\sqrt5y) + 4y^2 + 1 = 0$$ rotate the axes to eliminate the $xy$-term in the equation, then write the equation is standard form.
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Hyperbola with its directrix

The equation $9x^2 - 16y^2 -18x +32y-151=0$ represents a hyperbola . We have to find the equation of its directrix. I simplified the equation and got : $$(3x-1)^2 -(4y-1)^2 = 151$$ And found that ...
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Finding a mirror point on a parabola

What is the height of the ball at a point of 3 metres beyond where it was thrown, measured horizontally? How far is the ball from where it was thrown when its height has this value again? ...
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Geometric Invariants of a conic section

There are three independent invariants for every conic section, viz., $$[I_1,I_2,I_3]= [ (a + b + c), (a b -h^2), Det(( a,h,g), (h,b.f), (g,f,c) )] $$ How are they related to the known geometric ...
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Minimum Enclosing Ellipsoid To Maximal Enclosed Ellipsoid

Given a convex body $K$ and an ellipsoid of minimal volume which contains $K$, find the maximal ellipsoid contained in $K$. I have tried to multiply the matrix by 4 (since the eigenvalues are the ...