# Tagged Questions

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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### Find the latus rectum of the Parabola

Let $y=3x-8$ be the equation of tangent at the point $(7,13)$ lying on a parabola, whose focus is at $(-1,-1)$. Evaluate the length of the latus rectum of the parabola. I got this question in my ...
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### Graphing calculator leaving gaps in a drawn graph of a rotated parabola

I am graphing the equation of two rotated parabola on the graphing calculator and, after finding the y= form for each using the quadratic formula and entering them into a program to graph them they ...
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### Show that the tangent to the hyperbola $(x_0 , y_0)$ does not intersect the curve anywhere else.

Question: Consider the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ Given that the equation of the tangent at the point $(x_0 , y_0)$ is $\frac{xx_0}{a^2} - \frac{yy_0}{b^2} = 1$ ...
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### What region in $\mathbb{C}$ does $\left|{z-1}\right|+\left|{z+1}\right|$ = 2 describe?

I have played around with this a bit and keep getting something that doesn't seem right. Perhaps I'm overlooking something. Using the definition of distance in the complex plane I transform my ...
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### What is condition for second degree equation to represent a pair of straight lines?

According to my text the necessary and sufficient condition for a general equation of second degree i.e. $ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0$ to represent a pair of straight lines is that 1) the ...
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### How to calculate an inner ellipse points that is always a set distance from an outer ellipse points

I have an Ellipse with known coordinates , I would like to know how I can create an inner ellipse coordinates that are exactly 5 inches perpendicular from the outer ellipse points. Please see the ...
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### Why does this limit of hyperbolic cosines equate to a parabola?

I bolded my main question below, and I would like to understand why the following limit is true: $$\lim _{ n\rightarrow { 0 }^{ + } }{ \frac { \cosh { (nx) } -1 }{ \cosh { n } -1 } } = { x }^{ 2 }$$ ...
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### Plot the path of two axis relating to points on a super ellipse?

I have two axis linked to each other; and I need to move both axis 1 and axis 2 to follow the path of a known super ellipse - I have 38 points defined for the super ellipse. A picture is attached, I ...
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### Find the second focus of an ellipse given one focus, the point furthest from it, an arbitrary other point on the ellipse [duplicate]

I have the following three points: A: One focus of an ellipse C: The furthest point from this focus (the far side of the major axis) D: Some other arbitrary point on the ellipse From there, it's ...
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### Map an ellipsoid to a sphere

If I have a ellipsoid described by: $(\boldsymbol{x} - c)^T \boldsymbol{A} (\boldsymbol{x} - c) = 1$ How do I get the transformation to an unit sphere centered at the origin? From the principal ...
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### If 5 points are necessary to determine a conic, why are only 3 necessary to determine a parabola?

I've just been reading about how 5 points are necessary and sufficient to determine a conic section in Euclidean geometry (https://en.wikipedia.org/wiki/Five_points_determine_a_conic). But if ...
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### Find the radius of the circle for given conditions

A circle with center at origin passes through three points $P$, $Q$ and $R$ with the line segment $PQ$ as its diameter along $x$-axis. A line passes through $P$ intersects the chord $QR$ at point $D$. ...
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### Fit an ellipse to two points with a gradient condition on each point

For two Cartesian points $(x_1, y_1)$ and $(x_2, y_2)$ on an ellipse and two gradients $g_1$ and $g_2$ where the gradients describe the slope of the ellipse at each of the two points I want to ...
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### Is there an analytical way to fit a oblique ellipse cylinder to points?

For 2d ellipses this is a solved problem [1]. I've been searching the web for a while, but haven't come up with anything useful so far. So I think it is safe to ask here: Given a oblique ellipse ...
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### Identifying the conic given some conditions.

So I have to identify the conic which represents the centre of the circle which touches another circle externally, and also touches the x axis. Here's a link to the exact question with the equation ...
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### Ellipse construction question

This is a rough sketch to roughly explain the terms I will be using: Let's say you make a machine that would rotate the paper in some frequency $x$, let's say 1 rotation per second. The drawing ...
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### Finding the Cartesian equation of an ellipse (Midpoints)

Question: The normal to the ellipse $\frac{x^2}{25} + \frac{y^2}{9} = 1$ at a point $Q$ meets the coordinate axes at A and B respectively. As $Q$ varies, the locus of the midpoint of $AB$ is ...
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### find m so that this becomes the equation of an ellipse

For the equation $$(m − 2)x^2 + (y − 1)^2 − (m − 1)(m − 2) = 0 \textrm{ and } m \in \mathbb{R} \setminus \begin{Bmatrix}1, 2\end{Bmatrix}$$ Find $m$ so that this becomes the equation of an ellipse. ...
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### Best Fitting Pipe in parabolic trench

A work crew is digging a pipeline. The cross section of the trench is in the shape of the parabola $y = x^2$. The pipe has a circular cross section. If the pipe is too large, then the pipe will not ...
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### Deriving the equation of an ellipse from another related equation

Consider the equation for $x,y, \phi \in \mathbb{R}$ $$\left( \frac{x}{a} \right)^2 + \left( \frac{y}{b} \right)^2 - 2 \cos (\phi) \frac{xy}{ab} = \sin^2 (\phi)$$ It is supposed to be an elementary ...
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### How to interpolate elliptically

Given two orthogonal axes with different weightings along each axis, how do I interpolate elliptically between the two weightings? This is in 2d cartesian space. For example, axis1 might be Vector(2,...
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### What does “ b ~ length of semi-conjugate axis ” represents in the graph of hyperbola?

In the standard equation of hyperbola, $$\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$$ where $b^2=a^2(e^2-1)$ If i were to draw the graph of hyperbola what would it represent in the graph? As a represents ...
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### Rotation of conics [duplicate]

How to rotate a conic by an determined angle? Could someone give me the step by step? (I know how to rotate the coordinate system by that formula \begin{align} x &= x'\cos(a) - y'\sin(a) \\ y &...
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### How to get the distance in meters from the center of a parabolic image to a point?

I have a distance from the center of a 360° image to a particular point in pixels. I am trying to find a way to map that distance into dinstance in meters. The image is captured by a camera looking ...
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### Find the maximum up and down stroke of a piston as an elliptical plate revolves

Currently stuck on the following word problem (from Gersting, Technical Calculus and Analytical Geometry): An elliptical plate rotates on a shaft through its center. A pin 1 cm from the end of the ...
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### Area under a curve and its tangent [closed]

How can we calculate the area of the region bounded by the parabola $(y-2)^2$=$x-1$ , the tangent to the parabola at the point $(2,3)$ and the $x-axis$?
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### Solving Any Cubic

I am trying to show that solving any cubic can be done by intersecting a hyperbola with a parabola. I've tried doing so and substituting, but I continue to get stuck simplifying. I used the hyperbola ...
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### If The equation $ax^2+4xy+y^2+ax+3y+2=0$ represents a parabola then find the value of $a$.

Problem:If The equation $ax^2+4xy+y^2+ax+3y+2=0$ represents a parabola then find the value of $a$. My attempt-I known that in a parabola($e=1$)[where $e$ is eccentricity].So the distance of any ...
### Can every parabola be written in the form of a quadratic $y=ax^2+bx+c$ or $x=dy^2+ey+f$?
I understand that the graph of any equation of the form $y=ax^2+bx+c$ is a parabola (please correct me if I am mistaken). My question is about the converse: Can every parabola be written in the form ...