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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A ball is thrown upward from the top of a tower.If its height is described by $-t ^2+60t+700$,what is the greatest height the ball attains?

A ball is thrown upward from the top of a tower.If its height is described by $-t ^2+60t+700$,what is the greatest height the ball attains ? I've managed to get a solution by realizing that ...
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52 views

Ellipse Area (Trouble understanding answer)

Question: An elipse with equation $$ {x^2\over a^2} + {y^2\over b^2} = 1 $$ is enclosed by the hyperbolas given by $xy=1$ and $xy=-1$. , Determine the largest area of an ellipse enclosed ...
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1answer
51 views

Locus of vertex of a rectangle

If from the vertex of a parabola $y^2 = 4ax$ a pair of chords be drawn at right angles to one another and with these chords as adjacent sides a rectangle be constructed , then we have to find the ...
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2answers
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Find the tangent equation to the circle

The circle is given as $$x^2+y^2+z^2-7y+2z-8= 0$$ $$3x-2y+4z+3=0$$ at the point $(-3,5,4)$. I know the answer will be in the form of $$\frac{(x+3)}{l}=\frac{( y -5 )}{m}=\frac{( z-4)}{n}$$ but ...
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1answer
18 views

3 normals on a parabola

If $(x_1, y_1), (x_2, y_2)$ and $(x_3, y_3)$ be three points on the parabola $y^2 = 4ax$ and the normals at these points meet in a point then how will we prove that $$ \frac{x_1 -x_2}{y_3} + ...
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1answer
75 views

Chord of a parabola $y^{2}= 4ax$

Prove that on the axis of any parabola $y^2=4ax$ there is a certain point $K$ which has the property that,if a chord $PQ$ of the parabola be drawn through it ,then $$\frac{1}{PK^2}+\frac{1}{QK^2}$$ is ...
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1answer
127 views

The point of intersection of two perpendicular tangent lines to a parabola

If two perpendicular straight lines through the focus of the parabola $y^2 = 4ax$ meet its directrix in $T $ and $T'$ respectively. Show that the tangents to the parabola parallel to the ...
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2answers
45 views

Define ellipse and hyperbola in terms of distances from a point

Just as we say a circle is a locus of points that are equidistant from a single point. How to define an ellipse and a hyperbola in a similar way?
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1answer
175 views

Geometric derivation of the quadratic equation

The quadratic equation can be thought of as specifying distances in the Euclidean plane. It tells us that the $x$-intercepts of a function occur at a distance of $\frac{\sqrt{b^2-4ac}}{2a}$ from the ...
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0answers
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Proof of equation of ellipse

The ellipse can be defined as a conic section with eccentricity lesser than unity. How can you derive the equation of the ellipse using this definition? I can't find proof of the formula using this ...
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0answers
27 views

Polar form of generalized superellipse

I am looking for the polar form of the generalized superellipse: $$ \left|\frac{x}{a}\right|^{n_2}+\left|\frac{y}{b}\right|^{n_3}=1 $$ where $a$ and $b$ are the semi major and semi-minor axes. I have ...
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2answers
29 views

Given Function, find domain and description of graph $y = f(x)$

I am studying for Graduate Record Exam. The following question is difficult. Given the domain and description of $f(x) = 5 - (x + 20)^2$, including its shape, and the $x$ and $y$-intercepts To find ...
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0answers
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Construction new ellipse

Using a pencil the thread was pulled on the ellipse. Then the pencil started to rotate around the ellipse. How to prove that a new geometric figure which the pencil drew is also an ellipse (with the ...
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3answers
371 views

Need to find the ellipse of maximum area inscribed in a semicircle.

An ellipse inscribed in a fixed semi circle touches the semi-circular arc at two distinct points and also touches the bounding diameter. Its major axis is parallel to the bounding diameter. ...
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1answer
41 views

How would I put $x^2 + 4x + 25y^2 - 50y = -4$ into the equation for an ellipse?

The equation is $x^2 + 4x + 25y^2 - 50y = -4$. How would I put this into the equation for an ellipse $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$?
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3answers
20 views

Show that this section of the cone is a hyperbola

Show that section of a cone, with vertex at origin and base $x=a$ & $y^2+z^2=b^2$, intersected by a plane parallel to $XY$ axes is a hyperbola.
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3answers
67 views

What is the general equation of an ellipse that is not aligned with the axis?

I originally asked this in an answer to the following question: What is the equation of an ellipse that is not aligned with the axis?. As I noted in the opening paragraph I DO NOT HAVE THE NECESSARY ...
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1answer
61 views

Can't Simplify this equation for a Ellipse(Complex Numbers)

I'm asked to sketch the set $\{z \in C : |z + i| + |z + 1| = 2\}$. I've gotten to the point where I've got the modulus form of $|z + i| + |z + 1|$: $$\sqrt{x^2+(y+1)^2} + \sqrt{(x+1)^2+y^2} = 2$$ How ...
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1answer
21 views

Problem regarding parabola

While studying conic sections, in the parabola portion, I read that The sum of the ordinates of the extremities of the chords of the parabola $y^2=4.a.x$ which are parallel to each other is ...
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1answer
48 views

Parabola conic section

Two tangents to the parabola $y^2= 8x$ meet the tangent at its vertex in the points $P$ and $Q$. If $|PQ| = 4$, prove that the locus of the point of the intersection of the two tangents is $y^2 = 8 ...
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2answers
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Finding area of rectangle under a parabola asymmetrical with respect to the Y-axis: What did I do wrong?

I am using these as references: How to find the dimensions of a rectangle if its area is to be a maximum? Does the symmetry of a parabola in finding the maximum area of a rectangle under said ...
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1answer
48 views

Does the symmetry of a parabola in finding the maximum area of a rectangle under said parabola matter?

Apologies for my English, I'm a not a native speaker... So I've got this homework, about finding the maximum area of a rectangle under a parabola. I'm using this as a reference to do my work: How ...
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2answers
66 views

Determine the dimension of the set of surfaces of $\mathbb{P}^{3}$ that contain certain conic.

Let $C\subseteq\mathbb{P}^{3}$ be the conic of equations $$ C=V(X_{3}, X_{0}X_{2}-X_{1}^{2})=\{(t_{0}^{2}:t_{0}t_{1}:t_{1}^{2}:0)\in\mathbb{P}^{3}:(t_{0}:t_{1})\in\mathbb{P}^{1}\}. $$ I have to ...
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1answer
19 views

orthogonal diagonalization to sketch equation: $5x^2-24xy-5x=13$

to sketch this i wrote the equation down in the form: $X^TAX=13$ where $X^T=[x\:\:y]$ and $A=\begin{bmatrix} 5&-12\\ -12&-5 \end{bmatrix} $. Then, $A$ is orthogonally diagonalizable so there ...
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0answers
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Prove an ellipse is unique if the foci and a tangent are given.

Given 2 points $F_1$ and $F_2$ and a straight line $l$ which does not cross $[F_1F_2]$. Prove that there exists an unique ellips with $F_1, F_2$ as foci, and tangent $l$. What if $l$ crosses ...
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1answer
35 views

Finding the condition for point of intersection of three normals to a given parabola

Question: Suppose that the normals at three different points on the parabola $y^2=4x$ pass through the point (h,0). Show that h>2. My attempt: Equation of normal to parabola $y^2=4x$: ...
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1answer
25 views

Points of intersection of vector with cone.

I have a Vector $\vec A$ defined as : $(A_o+t*A_d)$ I also have a Cone with vertex (cone tip) V and axis direction $\vec D$, base radius R and height H. The cone angle can be computed via ...
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0answers
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How to find smallest tangent ellipse from multiple lines?

The ellipse I'm trying to calculate must be tangent to at least four lines. This ellipse must also intersect the other lines. I've tried using quadrilaterals and by transforming the equation of these ...
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40 views

Hyperbola and 3 normals from point P

From any point P on the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ three normals other than that at P are drawn. Find the locus of the centroid of the triangle formed by feet of the normals. Do we ...
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32 views

Is it actually important to translate the conics?

I've been studying conics and I'm curious about one aspect. There is a common way of doing it. Taking the quadratic form, we make a rotation and a translation and then the cross-term and the linear ...
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1answer
13 views

Eigenvalues order in conics reduction?

I'm doing exercises on the reductions of conics to canonic forms using eigenvalues. I'm trying to understand what does actually change when I put the two eigenvalues that I find in a different order ...
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0answers
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Inversion across an ellipse

Let's take an ellipse with the standard equation $$\frac{x^2}{9}+\frac{y^2}{4}=1$$ And I am trying to invert the following ellipse across that ellipse ...
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1answer
89 views

Locus of centres of circles tangent to two fixed circles?

Find the locus of the centres of circles tangent to two fixed circles. From my initial observations, I strongly think that the locus may be part of a hyperbola or some other conic? (because the ...
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0answers
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Inversion across a general ellipse

This paper is very useful in how it explains the mapping of any coordinates $(x,y)$ across an ellipse with the function $$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$$ to ...
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3answers
35 views

Check if disk lies within an ellipse

I have an ellipse in normal form centered at the origin and want to check whether a disk with given center point and radius is contained completely in the ellipse without touching it. If I could ...
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0answers
38 views

Reflective property of a hyperbola

An ellipse reflects an incident ray through one focus to the other as reflected ray and its special case of parabola likewise reflects rays parallel to symmetry axis after bouncing to go through its ...
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1answer
15 views

Coordinates of a string of beads

Given a number of circular beads of a given size strung on an ellipse with a known semi-minor axis, how can I calculate the position (let's say in Cartesian coordinates) of each bead's center? The ...
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1answer
28 views

Find one rational point on conics

We consider the equation: $Ax^2+Bxy+Cy^2+Dx+Ey-F=0$ with $A,B,C,D,E,F \in \mathbb{Q}$ If one has a rational point (a point whose coordinates are both rational) on the curve described by the ...
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2answers
31 views

Finding the major axis of an ellipse given the angle of a tangent

Given the angle between the tangent and the line that connects the point of tangency to each foci, and you are given the distance from one of the foci to the point of tangency. You are given two ...
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2answers
47 views

How to prove this theorem rhetorically?

It is not possible for a part of any of three conic sections to be an arc of a circle. It is asked to prove this theorem without using any notation or any modern form of symbolism whatsoever ...
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1answer
107 views

Volume of Revolution of a Transformed Ellipse?

I'm looking at an ellipse (a bunch of them actually) transformed by $h$ on the $x$-axis away from the center and rotated by an angle of $Q$ from the $xy$ axis. I got the following equation: the $x$ ...
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2answers
28 views

Tangents and Normals of rectangular hyperbolas

Please, could someone explain the solution to (d) I solved (a), (b) and (c) however, I don't understand how to calculate (d). (a) displayed in the question (b) $ q^2 y + x = 10q $ (c) displayed in ...
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1answer
26 views

Coordinate of $S(s,t)$ for Which Area of Quadrilateral is Maximum.

Let $P(-2,3)\;\;,Q(-1,1)\;\;,R()$ and $S(2,7)$ be $4$ points in order on the parabola $y=ax^2+bx+c$.Then the coordinate of $R(s,t)$ such that the area of Quadrilateral $PQRS$ has maximum ...
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1answer
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Distance between feet of perpendiculars from focii of ellipse

$$Tangent\quad drawn\quad to\quad ellipse\quad { x }^{ 2 }+{ 2y }^{ 2 }=6\quad at\quad point\quad (2,1).If\quad A\\ and\quad B\quad are\quad the\quad feet\quad of\quad pependiculars\quad from\quad ...
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1answer
39 views

Points on a normal to a superellipse at distance $d$ from the curve

Given a point P0 $(x0, y0)$ lying on a super ellipse, $(x/a)^n + (y/b)^n = 1$, where $2 <= n <= 5$, I'm trying to derive an equation to describe the point P1 (x1, y1) lying on the normal through ...
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1answer
44 views

What's the importance of conics and quadrics in the context of a course of pure mathematics?

I've studied conics and quadrics in the past (specifically, in a course of analytic geometry). For these courses, we usually learn the basics of linear algebra and we apply these to conics and ...
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2answers
58 views

Find $a$, $b$ such that the ellipse $(x/a)^2 + (y/b)^2 = 1$ passes through $(\sqrt 2, 2)$ and has minimum area

I am working on a problem in which, for $a$, $b \gt 0$, we let $(x/a)^2 + (y/b)^2 = 1$ describe an ellipse. I am required to use the method of Lagrange multipliers and the corresponding second ...
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1answer
73 views

why there are no parabolic (on a paraboloid) non-euclidean geometry?

I have seen in many contexts that Euclidean geometry is called also "parabolic geometry". As in many things in mathematics (conics, differential equations, algebraic equations) the terms: ...
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34 views

Rewrite the equation of a conic in cartesian coordinates

Consider the equation for a conic in polar co-ordinates $(r,\theta)$ $$r = \frac{k}{1 - e\cos(\theta)} \qquad \qquad (1)$$ in the case where $k > 0$ and $e > 1$. Show that ...
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1answer
18 views

Find the width of an ellipse, on given Y position.

Look at picture below : Q: I wonder what's the equation to find $L$ (the red line on {x',y'}, which x' will always be $0$), ...