Questions tagged [conic-sections]

For questions about circles, ellipses, hyperbolas, and parabolas. These curves are the result of intersecting a cone with a plane.

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34 views

Proof of why conics map to conics after a perspective transformation

Background Consider a world where the ground is the standard $x$-$y$ plane with a Cartesian grid on it. The graph of a parabola $x^2 = 4ay$ is on this $x$-$y$ plane. A person of with eye-level $h$ ...
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Geometric proof for why the midpoints of parallel chords of a parabola lie on the same line parallel to the axis

I was trying to figure out a geometric proof for why the midpoints of parallel chords of a parabola lie on the same line which is parallel to its axis. I searched on StackExchange and people have ...
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Geometric proof that the product of the $x$-intercepts equals the $y$-intercept for a monic quadratic

I know you can prove that the product of the roots of the monic quadratic $x^2+a_1x+a_0$ equals the $y$-intercept $a_0$ by comparing its coefficients to the coefficients of $(x-m)(x-c)$ where $m$ and $...
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($y=ax^2+bx$) finding range of values for $a$ and $b$ so that the parabola's height is always greater than the distance between its zeros

So long story short, teaching myself algebraic and graphical modelling because of covid. I can't seem to find anything online about my particular question. Let's say an object is launched from the ...
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2answers
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Ellipse general equation from dimensions, offset, and tilt angle

Given an ellipse with the following parameters: $a$ = semimajor axis $b$ = semiminor axis $\theta$ = tilt angle from horizontal $(\Delta x, \Delta y)$ = position of the center How do I find the ...
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1answer
95 views

What kind of curve do these lines make?

Our teacher has given us a question to be solved. What curve does the intersection points of the given lines make? A parabola, hyperbola, or none of them? (please look at the image I've posted. I am ...
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why apollonius defines focus of central conics as points S,S' such that AS.S'A are “one-fourth part of the figure of the figure of the conic”?

I am trying to understand this part of apollonius conics by heath, page 112 The foci are not spoken of by Apollonius under any equivalent of that name, but they are determined as the two points on ...
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1answer
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Violating Pythagorean theorem

In an ellipse, distance from the center to one of the vertex $(v_1)$ is $a$; center to one of the co-vertex is $b$ and $c$ is the distance from the center to the focus that is close to the vertex $(...
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1answer
32 views

Packing parabola with the chain of internally tangent circles

Related to the question Condition for perfect packing of ellipse with circles along the major axis An attempt of "perfectly pack" a parabola $x=ay^2$ as a special king of ellipse with ...
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1answer
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Derivative of A Quadratic Form with a Change of Basis

The derivative of a quadratic form is given by: $\Delta_{X} X^{T}AX = 2AX$ where A is symmetric. What is the derivative of the following: $\Delta_{X} (DX)^{T}A(DX)$, where $A$ is positive and ...
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Finding position of endpoint on an ellipse's perimeter given axes

Given the semiaxes and $d$ (where $d$ starts at the semimajor axis and ends at point $A$), is there a formula to calculate the position of the endpoint on the ellipse perimeter? In my case, the ...
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Angular coefficient of the straight line tangent to the parabola at one of its points: alternative proof

Let be $P_0(x_0,y_0)\in\mathcal P : \, y=ax^2+bx+c$. We know that the $m$ angular coefficient of the straight line tangent $t$ in $P_0$ is: $$m=2ax_0+b \tag 1$$ To find the $(1)$ I before consider the ...
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3answers
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How to prove distance from foci on an ellipse is equal to twice the semi-major axis (for specific ellipse)

Prove that for any point (x,y) on the conic, the sum of the distances to the two foci is always twice the semi-major axis. I know that this can be proven in general for all ellipses but the practice ...
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1answer
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Tangent to an ellipse

I want to find the tangent on point $(x_0, y_0)$ to an ellipse with equation $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$. We assume $y_0$ is positive. We can derive $y=\frac b a \sqrt{a^2-x^2}$. In order to ...
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A bound on the maximal singular value of a linear map which preserves an ellipse

Let $0<a<b$, $ab=1$, and let $$ D_{a,b}=\biggl\{(x,y) \,\biggm | \, \frac{x^2}{a^2} + \frac{y^2}{b^2} \le 1 \biggr\} $$ be the ellipse with diameters $a,b$. Let $A \in \operatorname{SL}_2(\...
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Derivation of equation of hyperbola - Where does the b²=c²-a² come from? [duplicate]

I'm learning about conic sections as defined using focci. Every derivation I find for the hyperbola get's this definition out of the blue: $b²=c²-a²$ It totally looks like a triangle somewhere, but ...
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1answer
52 views

Singular values of matrices which preserve the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} \le 1$

Let $0<a<b$, $ab=1$, and let $$ D_{a,b}=\biggl\{(x,y) \,\biggm | \, \frac{x^2}{a^2} + \frac{y^2}{b^2} \le 1 \biggr\} $$ be the ellipse with diameters $a,b$. Let $A \in \operatorname{SL}_2(\...
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1answer
82 views

Homographic function: alternative proofs to obtain $ad-bc$

Considering the function, $$y=\frac{ax+b}{cx+d}\tag1$$ If $c = 0 \wedge d\neq 0$, the function represents a straight line of equation $$y=\frac ad x+ \frac bd$$ If $c ≠ 0$ and $ad = bc$ the function ...
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2answers
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coordinates of focus of parabola

Find the coordinates of focus of parabola $$\left(y-x\right)^{2}=16\left(x+y\right)$$ rewriting: $(\frac{x-y}{\sqrt{2}})^2=8\sqrt2(\frac{x+y}{\sqrt{2}})$ comparing with $Y^2=4aX$ $4a=8\sqrt2,a=2\sqrt2 ...
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1answer
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Line integral of an ellipse using Green's theorem

Let C be the curve in $\mathbb{R}^2$ defined by, C: $\dfrac{x^2}{9}+\dfrac{y^2}{4}=1, x\geq0,y\geq0$ Compute the line integral, in a direction so that the $y$-coordinate increases: $\int_C (2x-3y)dx+(...
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Given ellipse of axes $a$ and $b$, find axes of tangential and concentric ellipse at angle $t$

Let’s say I have an ellipse with horizontal axis $a$ and vertical axis $b$, centered at $(0,0)$. I want to compute $a’$ and $b’$ of a smaller ellipse centered at $(0,0)$, with the axes rotated by some ...
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1answer
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Find the eccentricity of the conic $4x^2+y^2+ax+by+c=0$, if it tangent to the $x$ axis at the origin and passes through $(-1,2)$

Solving this would require three equations (1) Tangent to x axis at origin Substituting zeroes in all $x$ and $y$ gives $c=0$ (2) Passes through (-1,2) $$4(1)+4-a+2b=0$$ $$-a+2b=-8$$ How do I find the ...
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geometric derivation of formulas for conic sections from a double cone

I'm trying to find derivations for the ellipse and the hyperbola that use geometric proofs such as the one found for the parabola on wikipedia (image for clarity) I'm aware that there are proofs ...
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1answer
46 views

How do I find the point on a given hyperbola that passes through a given point not on the hyperbola?

There's a family of lines and where they intersect, the envelope, is a hyperbola. The hyperbola's center is at the origin and the $y$-axis is its axis of symmetry, and I know both $a$ and $b$. I'm ...
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The mirror point of an ellipse's focus about the tangent line through a point is collinear to said point and the other focus

I was reading a bit about ellipses, specifically why light sent out from one focus will reflect off the ellipse and converge back to the other focus. The proof I was reading involves a construction as ...
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How to find the point of intersection of normals to a general conic drawn from points where tangents from an external point P meet the conic?

Let's say we are given a general conic's eqn: $ax^2+by^2+2gx+2fy+2hxy+c=0$, and also a fixed point $P(r,s)$. Now I draw a pair of tangents to the conic from $P$. Let these tangents meet the conic at ...
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1answer
84 views

3 new Points lying on Jerabek Hyperbola?

R is the circumcenter, H is the orthocenter of the triangle ABC. Then points F,G,E are the points of intersection of the altitudes with the sides of the triangle ABC. U, V, W are the intersection ...
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3answers
110 views

Triangle tangent to 3 Parabolas, finding the common area

Triangle $ABC$, $AB=4$, $BC=15$, $AC=13$. Two sides are tangents to the respective Parabolas. We have to find the area shaded. My approach- I tried finding the area of the quadratures(Archimedes) ...
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1answer
37 views

Area between parabola and a line that don't intersect? 0 or infinity

Came across a problem on social media, Find the area of the region bounded by a parabola, $y = x^2 + 6$ and line a line $y = 2x + 1$. I tried to draw it on paper and they didn't seem to intersect. ...
3
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2answers
76 views

Ellipse on complex plane

I have found that the curve $z(x)$ on the complex plane with $$ z=\frac{ax+b}{x^2+1} $$ at real $x$ spanning from $-\infty$ to $+\infty$ looks very similar to an ellipse at any complex parameters $a,b$...
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1answer
47 views

The locus of vertex of a parabola, given that an orthogonal intersection is made with another having specified latus rectum and orientation.

Full Question: A variable parabola of fixed latus rectum 4b and having axis parallel to x–axis, lies completely in Ist and IVth quadrant and cuts the fixed parabola $y^2=4ax$ orthogonally. The locus ...
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Find the polar equation of the ellipse

Find the polar equation of the ellipse with eccentricity e and semi major axis a, considering one focus of the ellipse at origin and the corresponding directrix to the right of the origin. Also, if $a=...
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2answers
52 views

Derive equation $SS_1-T^2=0$ for pair of tangents to an ellipse

I have read the formula for the pair of tangents from $(x_1,y_1)$ to the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ is $$SS_1-T^2=0$$ where $S :\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$,$S_1:\frac{x_1^2}{a^...
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1answer
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Finding the intersection of two cones (modeling an inflatable structure formed from truncated cones)

I am trying to model the join of the leading edge of this 'wing'. The tubes are inflated, each section is conical in shape. I know the radius of the tubes at each joint. Given a vector in 3D space of ...
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2answers
68 views

Orthocenter of triangle with vertices on the hyperbola is also on the hyperbola

The points $P(p, 1/p), \,Q(q,1/q), \,R(r, 1/r)$ and $S(s, 1/s)$ lie on the curve $xy = 1$. a) If $PQ || RS$, show that $pq = rs$. b) Show that $PQ\perp RS$ if and only if $pqrs=-1$. c) Use part b to ...
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1answer
39 views

How to find the required equation for a family of curves?

The question given is: Tangents are drawn from two points $(x_1,y_1)$ and $(x_2,y_2)$ to $xy=c^2$. The conic passing through the two points and through the four points of contact will be a circle, ...
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0answers
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Proof that in hyperbola sum of squares of a and b is equal to c. [duplicate]

Where c is the focus of the hyperbola.I came across this problem when I was going through the derivation of equation of hyperbola. A substitution was made there and difference of squares of c and a ...
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3answers
98 views

Prove that the maximum area of a rectangle inscribed in an ellipse is $2ab$

Prove that the maximum area of a rectangle inscribed in an ellipse $\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$ is $2ab$. My attempt: Equation of ellipse: $\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$. Assume that $...
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0answers
43 views

Importance of the mathematical differentiating of an ellipse (or of a conic)?

Supposing to have a canonical ellipse (or other conics, parabola, circle, or hyperbola): $$\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1, \quad \text{or} \quad \frac{(x-x_0)^2}{a^{2}}+\frac{(y-y_0)^2}{b^{...
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Transformation of ellipses into lines

Suppose I have a collection of ellipses defined in the first quadrant (i.e. the points $(x, y)$ with $x \geq 0, y \geq 0$) as such $\left( \frac{x}{a_i}\right)^2 + \left( \frac{y}{b_i}\right)^2 = 1$, ...
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2answers
157 views

A locus problem related to circumcenters and conic sections

Given a point $A$, a circle $O$ and conic section $e$, if $BC$ is a moving chord of the circle $O$ tangent to $e$, then prove that the locus of △$ABC$'s circumcenters $T$ is a conic section. The ...
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Parametrizing any arbitrary Conic

Given the implicit equation of a Conic $C$, how to determine its parametric representation? I went thought this report and it has an algorithm: Fix a point $p$ on the conic. Consider the pencil of ...
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1answer
124 views

Locus of a point with constant distance ratio $e$ to two circles.

Please help to obtain.. in as elegant a form as possible.. the locus of point $P$ equations if its distances to two circles: $$ (x-h)^2 + y^2 = a^2;\;(x+h)^2 + y^2 = b^2 ;\;$$ are in a constant ratio $...
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57 views

Solving two equations of ellipses simultaneously

Ellipse equation: $$\frac{(x-h)^2}{a^2}+\frac{(y+k)^2}{b^2}=1 $$ The radii $(rx, ry)$ is given, which value $a$ and $b$ are known constants. Another piece of useful information is the coordinates of ...
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1answer
50 views

Find side length of square with vertices on line $y=x+8$ and parabola $y=x^2$

Let ABCD be a square with the side AB lying on the line $y=x+8$. Suppose C, D lie on the parabola $x^2=y$. Find the possible values of the length of the side of the square. I'm not sure how to start, ...
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1answer
18 views

Proof involving showing the angles created by a parabola, a tangent, a line parallel to the axis of parabola, and the parabola's focus are equal.

I have to prove that angle $ \alpha $ is equal to angle $ \beta $. The book suggested to show that an isosceles triangle exists. I was able to show that $FP$ = y + p ,where p is the directed distance ...
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1answer
21 views

Centre and axes of symmetry for quadrics

Since it is very hard to find good and organized scripts online for quadrics I came here to maybe find some answers to questions that've been bugging me lately: If a quadric has two axes of symmetry, ...
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1answer
39 views

How to find the general form of minimum distance from the point (m,n) to the ellipse by using Lagrange Multiplier?

Excuse me! I have tried to solve this problem for a long time but I have stuck in this step. This is my work picture01. This is my work picture02. I cannot simplify y in term of a b m and n. It’s too ...
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1answer
21 views

Plotting the Vertices of a Rotated Ellipse with Non-Origin Centre (MATLAB)

I'm trying to plot the vertices of an ellipse of the form: $Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$. Here's my attempt: ...
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1answer
39 views

Affine Linear Transformations of conics (parabola.)

I want to apply an affine-linear Transformation $L:\mathbb{R}^2 \rightarrow \mathbb{R}^2: x \rightarrow Lx+b$ on a parabola e.g. $y=x^2$. So I interpret the parabola as a conic and represent it as $\...

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