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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What is the significance of Latus Rectum?

So I just completed the chapter Conic Sections and the one thing I could not understand is what is the use of Latus Rectum? It is defined as " Line segment passing through the focus and parallel to ...
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Question related to shifted parabola

I have problem in dealing the question related to the equation of shifted parabola. I have a question as "A parabola whose latus rectum is $4c$, slide between two rectangular axes. Find the locus of ...
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Line $mx + ny = 3$ is normal to the hyperbola $x^2 – y^2 = 1$

If the line $mx + ny = 3$ is normal to the hyperbola $x^2 – y^2 = 1$, then evaluate $\frac{1}{m^2}+\frac{1}{n^2}$. I compared given equation of normal to equation of normal at parametric point i.e $x\...
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132 views

Intersection of two ellipses

I do not have a background in mathematics and geometry, so I will be so thankful if someone please give me a simple way to get the following, as I tried to understand some posts related to this issues,...
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2answers
60 views

Using a parabola for interpolation

I am trying to use a parabola to interpolate between 3 values and I have been struggling with finding an equation that works for me. The constraints are: passes through the points $(0, s), (x,m), (...
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1answer
51 views

To find the locus of vertices of shifted parabola [closed]

How to deal with this question. Please help. "Parabolas are drawn to touch two given rectangular axes and their foci are all at a constant distance $c$ from the origin. Find the locus of the vertices ...
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3answers
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Show that the equation of the tangent to the parabola $y^2=4ax$ at the point (p,q) is $qy=2a(x+p)$

Question: Show that the equation of the tangent to the parabola $y^2=4ax$ at the point (p,q) is $qy=2a(x+p)$ These are my two approaches: First approach: If we have $(p,q)$ as $(x_1,y_1)$ $$y^...
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4answers
312 views

Finding Maximum Area of a Rectangle in an Ellipse [duplicate]

Question: A rectangle and an ellipse are both centred at $(0,0)$. The vertices of the rectangle are concurrent with the ellipse as shown Prove that the maximum possible area of the ...
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3answers
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Why is the graph of a quadratic function a parabola?

I'm sorry for the stupid question, but it seems that extensive googling didn't yield an answer. I've learned about parabolas, and how the parabola is the curve that is equidistant from a point (Focus)...
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1answer
30 views

Taylor Polynomium and Conic Section

A real function of two variable is given by: $ f(x,y)=exp(x+y)·cos(x-y) $ The approximating polynomium of 2nd degree for f(x,y) with converging point $(x_0,y_0)=(0,0)$ called $P_2(x,y)$ a) Find $P_2(...
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Area covered by fixed perimeter around polygon.

Suppose I have a polygonal field with a post at each vertex and a non-extensible rope threaded through each post around the perimeter but with some slack. How can I determine the perimeter of the area ...
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Why can the equation of the tangent can be obtained by replacing $x$ with $x_0$ and similarly for $y$?

The non-rotated ellipse centered at the origin has equation $$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$$ We can show via implicit differentiation that the equation of the tangent at $\left (x_0,y_0 \right )$...
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1answer
44 views

Finding foci and vertices of an ellipse, know if the foci are located on the y axis or not given equation

When doing a problem where you have to find the foci and vertices of an ellipse given and equation like $9x^2-36x+4y^2=0$, The answer will change based on if they are or not, so how do you tell from ...
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2answers
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How to find an ellipse given a set of $x$ and $y$ values?

I have the function $f(x,y) = ax^2 + bxy + cy^2 + dx + ey + 1 = 0$, and have a set of 10 $x$ values and corresponding $y$ values where $x = (x_1, x_2, ... x_{10})$ $y = (y_1, y_2, ... y_{10})$ And ...
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0answers
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Finding foci, asymptotes, and a vertices of a hyperbola given an equation

I'm given the equation $4x^2-y^2-24x-6y+23$ and asked to find the foci, vertices and asymptotes. The book showed me how to do it given an equation in the form of $(x^2/a)-(y^2/b)=1$, but didn't show ...
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1answer
18 views

Limiting points subtend right angle at the centre

If the limiting points of the system of circles $x^2+ y^2+ 2gx +w(x^2+ y^2+ 2fy + k)=0$ where $w$ is a parameter , subtends a right angle at origin then find value $k/f^2$? I know that limiting point ...
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135 views

Possibly rotated parabola from three points

I'm looking for a possibly rotated parabola in the plane, i.e., the solution to a quadric like $$ Ax^2 + 2Bxy + Cy^2 + Dx + Ey + F = 0 $$ where exactly one of the eigenvalues of $$ \begin{bmatrix} A &...
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1answer
42 views

Confusion on polar coordinates of an ellipse

The polar coordinates of an ellipse are given by: $$x=\frac{abcos(\theta)}{\sqrt{b^2cos^2(\theta)+a^2sin^2(\theta)}}$$ $$y=\frac{absin(\theta)}{\sqrt{b^2cos^2(\theta)+a^2sin^2(\theta)}}$$ However, I ...
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Universal Parabolic Constant

I recently learned of a constant that arises in parabolas, similar to that of $\pi$ for circles. Like $\pi$ being the ratio of the circumference of the circle to its diameter, this constant $\sqrt2+\...
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1answer
44 views

Relation between $a$ and $b$ when equation of obtuse angle bisector is $ax+by-3=0$

The combined equation of bisector of angles between the lines $L_1$ and $L_2$ is $$2x^2-3xy-2y^2-x+7y-3=0$$ $P(4,-3)$ is a point on $L_1$. If the equation of obtuse angle bisector is $ax+...
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1answer
69 views

How to prove that the ellipse is a periodic orbit knowing that the orbital derivative of a function V is zero on there

The question is as follows: Show that the orbital derivative of the function $V=(1-x^2-2y^2)^2$ is zero on the ellipse $x^2+2y^2=1$, and explain why you can deduce that the ellipse is a periodic ...
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1answer
46 views

Determining the normal of an ellipse

Given I have (in a 2D coordinate system) an ellipse with the center at $(c_x,c_y) = (0,0)$ where I do not know the actual value of the major an minor axis but I have the ratio $r=\frac{a}{b}$ and an ...
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1answer
44 views

Is there a name for this quantity which is similar to the focus of a parabola?

Suppose we have parabola $y=ax^2+bx+c$, which has focus at $(-\frac b{2a},\frac 1{4a}-\frac {b^2}{4a}+c)$. There is a line $\ell$ at $y=\frac{a^2-b^2}{4a}+c$ which has the following property: any ...
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1answer
37 views

length of a tangent

The two tangents to a circle are represented by $2x^2-3xy+y^2=0$ . A circle of radius=3 is in first quadrant . "A" is a point of tangency where one of these lines meet.What is length OA where $O$ is ...
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1answer
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$y^2 = 2a(x+a\sin \frac{x}{a})$ and tangents parallel to $x$ axis

Prove that all the points on the curve $$y^2 = 2a(x+a\sin \frac{x}{a})$$ at which tangent is parallel to the axis of $x$, lie on a parabola. Here slope of tangent at $(h,k)$ must be $0$. After ...
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2answers
101 views

Proving that the line joining $(at_1^2,2at_1),(at_2^2,2at_2)$ passes through a fixed point based on given conditions on $t_1,t_2$

Problem:If $t_1$ and $t_2$ are roots of the equation $t^2+kt+1=0$ , where $k$ is an arbitrary constant. Then prove that the line joining the points $(at_1^2,2at_1),(at_2^2,2at_2)$ always passes ...
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1answer
73 views

Explicit formula for conformal map from ellipse to unit disc (interior to interior)

I was originally looking for a conformal map that maps a punctured unit disc to unit disc. The only answer I can find lead to this resource. The final step of the answer given rely on a conformal map ...
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2answers
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tangent of an ellipsis $c^2=a^2m^2 + b^2$?

If $y=mx + c$ is a tangent to an ellipsis $(\frac{x^2}{a^2})+(\frac{y^2}{b^2})=1$ Show that $c^2=a^2m^2 + b^2$. So for this question, first off I tried to differentiate it using implicit ...
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3answers
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ellipse polar co-ordinate conversion

I have a somewhat trivial question out of interest. Given the equation of an ellipse $$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$ why is the substitution $x = \sqrt{a}\cos t$ and $y = \sqrt{b}\sin t$ ...
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What is the meaning of the locus of points P satisfying some conditions?

A rod AB of length 15 cm rests in between two coordinate axes in such a way that the end point A lies on x axis and end point B lies on y axis. A point P(x,y) is taken on the rod in such a way that AP ...
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2answers
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How to find the equation of circle that passes through ($5,3$) , ($7,-2$) and ($-4,4$) circle with center at origin ($0,0$) and radius $r$?

It is a challenge assignment on our class and I can't figure out how to solve it I always got stuck it is not the same as the other examples which are easy to solve. thanks
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1answer
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What do I get if my Directrix and Focus are the same?

In my calculus class today we had a discussion about whether you'd get a vertical line or just a point when your directrix and focus become the same point. What would happen?
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Why is the focus of the parabola not within the parabola in the following result?

So i'm going through my book and try to solve the following question: Find the equation of the parabola which is symmetric about the y axis and passed through the point (2,-3). Since it passes ...
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4answers
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I do not get this question at all. I need to prove the an equation has a minimum. Quadratics involved.

Prove that $f(x)= (x-a)^2+(x-b)^2$ has a minimum when $x= \frac{a+b}{2}$. (Prove not verify) I do not get this question whatsoever, please help me.
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0answers
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angle between hrizontal and a line connecting the center of an oblate ellipse to a point in space

I would like to know how I can calculate the angle $\alpha$ in an oblate ellipse similarly to the sphere.
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1answer
62 views

Construct ellipse from two arbitrary points and a given focal point

Can an ellipse be constructed from these three given points: Focal point $\mathrm F$ Two arbitrary points $\mathrm U$, $\mathrm V$ lying on the ellipse The background is a orbital maneuver ...
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1answer
38 views

Conics and Loci Question (Hyperbolae and Circles)

A circle has the equation $x^2 + y^2 = r^2$. Tangents are drawn from a point $P(x_1,y_1)$ to the circle and these touch the circle at points $A$ and $B$. If the position of $P$ can vary and the locus ...
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2answers
24 views

How to Change an equation into Ellipse Form

I know how to arrange a normal equation into an ellipse form, but this one is slightly different. $x^2+2xy+5y^2=1$ Any help with this would be greatly appreciated. Thanks
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1answer
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General form to standard form regarding ellipse?

I've tried 2 hours to do this so I hope someone can help me: $$11400000=-0.64x^2+2560x-y^2+6000y$$ It says that it have to equal an ellipse with center at the point $(2000,3000)$ and a horizontal ...
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1answer
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Points with constant polar w.r.t to a tangent conic bundle

Consider the conic bundle of $\mathbb{P}^2(\mathbb R)$with matrix $$A(\lambda,\mu)=\begin{pmatrix} 0 & \mu & \mu \\ \mu & 0 & \lambda \\ \mu & \lambda & 0 \end{pmatrix}$$ This ...
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1answer
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Is the Locus circle?

Locus of points such that sum of it's distances of them from four fixed points remains constant? Is the locus circle? I was not able to solve it as there were four radicals. Is it a theorem?
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A question an normal to the circle

The equation of the normal to the circle $(x-1)^2+(y-2)^2=4$ which is at a maximum distance from the point $(-1,-1)$ is (A) $x+2y=5$ (B) $2x+y=4$ (C) $3x+2y=7$ (D) $2x+3y=8$ Since its a normal to ...
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2answers
148 views

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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1answer
34 views

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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3answers
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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: $$\frac{xx_0}{a^2} + \frac{yy_0}{b^2} = 1$$ I've ...
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0answers
16 views

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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30 views

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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42 views

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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49 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 $$C: (x-a)^2+y^2=r^2$$...
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1answer
37 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 ...