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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Intersection of cone and cylinder layout formula for sheet metal application

A common part in HVAC is a cylindrical pipe intersecting a truncated cone. I am designing a machine to mass produce this part. I would cut the parts out of sheet metal and roll them up to form the ...
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How do you find $∠XPC$ + $∠XPB$ such that $PB+PC$ is maximum where $P$ is a point on $f(x) = (x-1)(x-3)(x-5)$?

Problem: $f(x) = (x-1)(x-3)(x-5)$ intersects the x axis at $A(1,0)$, $B(3,0)$ and $C(5,0)$. A point $P(t,f(t))$ is selected on the curve such that $PB+PC$ is maximum and $t \in (3,5).$ Let $PX$ be ...
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How to derive the equation of tangent to an arbitrarily point on a ellipse?

Show that the equation of a tangent in a point $P\left(x_0, y_0\right)$ on the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$, could be written as: $$\frac{xx_0}{a^2} + \frac{yy_0}{b^2} = 1$$ ...
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Eigenvalue counting function for the Dirichlet and Neumann Laplacians on a rectangle

I'm doing some research on the Eigenvalue counting function for the Dirichlet and Neumann Laplacians on a rectangle and I was wondering if anyone knew why we consider the integer lattice points within ...
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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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30 views

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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30 views

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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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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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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88 views

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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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
59 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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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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2answers
37 views

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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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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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
19 views

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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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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1answer
25 views

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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1answer
16 views

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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1answer
26 views

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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1answer
59 views

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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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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1answer
28 views

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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2answers
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Number of inscribed triangles in a rectangular hyperbola touching a parabola [on hold]

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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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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1answer
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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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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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2answers
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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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1answer
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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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32 views

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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39 views

A circle pass through origin and centre is $(3,-3)$ and find coordinated point on the circle [closed]

A circle pass through origin and centre is $(3,-3)$ and line $y=x-6$ meet the circle at point $P$ and $Q$. Find coordinated of point on the circle where tangent are parallel to line $PQ$. I got the ...
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Directrix and foci relationship on a hyperbola

on a parabola the distance from the vertex is equal to the distance from the directrix. Is this the case with hyperbolas? I have looked on multiple math websites and they don't state this but from ...
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2answers
63 views

Find the eccentricity of a conic

Find the eccentricity $e$ of the conic $$S \equiv 39x^2+11y^2-96xy+14x+2y-34=0.$$ My try: Comparing with general second degree conic $$ax^2+2hxy+by^2+2gx+2fy+c=0$$ we have $a=39$, $b=11$, $2h=-96$, ...
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3answers
34 views

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 ...
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Help with a geometrical problem on ellipses

Let $E$ the ecllipse, with center $O = (0,0)$, focus $F=(4,0)$ and vertex $V=(5, 0)$. Let $N$ be a point on the ellipse $E$, and let $Q$ be the orthogonal projection of $N$ onto the y-axis. Find the ...
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42 views

Volume bounded between an Ellipsoid and a Cone?

I'm a bit confused about how I would be able to find the volume bounded by a cone of known theta and an oblate spheroid of b = c. I'm trying to use triple integrals for the solution, and I think I ...
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21 views

Geometrical place of a point R

Given the ellipse E $4x²+9y²=36$ and a point $P=(4,7)$. Let $Q=(x,y)$ point of the ellipse and $R$ a simetric point of Q respect $P$. Find the geometrical place of $R$. Ok, i think that R belong to a ...
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1answer
14 views

Number of common chords of the two parabolas

How can we find common chords to the parabolas $$ (y-2)=(x-3)^2$$ and $$(x-2)=(y-3)^2$$ without drawing graphs. What i have done is i have subtracted both of them and i got $$(y-x)=(x-y)(x+y-6)$$ ...
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2answers
28 views

Maximum area of $\Delta QSR$

The circle $C \equiv x^2+y^2=1$ cuts $X$ and $Y$ axes at $P$ and $Q$ Respectively. if another circle with centre $Q$ and variable radius is drawn so that it meets $C$ at $R$ and the line $PQ$ at $S$. ...
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1answer
41 views

Area of Triangle in ellipse

Full question: Prove that the area of the triangle formed by three points of an ellipse, whose eccentric angles are $\theta , \phi$ and $\psi$ , is ...
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1answer
26 views

Find the equation of the parabola with focus at $(-1,0)$ and vertex at $(3,0)$

For this problem I cannot figure out how to find the directix, defined to be perpendicular to the axis of symmetry. The general point $(x,y)$ on parabola needs to be far from $(-1,0)$ as it is from ...
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21 views

Hyperbola: A case of an ellipse?

Can i treat a hyperbola as a special case of ellipse. Like substituting b^2 with -b^2 Would all things still work? And also, why is a parabola different from the family of (circle, ellipse, ...
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Diagonal of parallelepiped circumscribed around ellipsoid is constant

There are many rectangular parallelepipeds that can be circumscribed around a given ellipsoid in $\mathbb R^n$. Prove that the length of the main diagonal does not depend on the choice of such ...
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Find equation of circle in the first quadrant touches $x$-axis $y$-axis and straight line $3x-4y-20=0$ . The point $H(12,4)$ lies on the straight line

1)Find equation of circle 2)Equation of another tangent from point $H$ to the circle The circle in the first quadrant touches $x$-axis $y$-axis and straight line $3x-4y-20=0$. The point $H(12,4)$ ...
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4answers
32 views

Polar equation of an ellipse given the origin coordinates and major and minor axis lengths?

I've been trying to create a polar equation that will give me all points on an ellipse with the independent variable being theta and the dependent variable being the radius, but I'm having a great ...