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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5
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4answers
231 views

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 ...
0
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0answers
28 views

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 ...
1
vote
3answers
35 views

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 $\frac{xx_0}{a^2} - \frac{yy_0}{b^2} = 1$ ...
0
votes
4answers
71 views

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 ...
1
vote
2answers
76 views

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 ...
0
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1answer
39 views

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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3answers
31 views

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: $$\lim _{ n\rightarrow { 0 }^{ + } }{ \frac { \cosh { (nx) } -1 }{ \cosh { n } -1 } } = { x }^{ 2 }$$ ...
0
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0answers
14 views

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 ...
1
vote
1answer
74 views

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 ...
0
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1answer
36 views

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 ...
2
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0answers
31 views

Finding a curve that is orthogonal at $(1,1)$ to the set of given parabolas

I am given a DE like the following: $$\frac{dy}{dx} = \frac{2xy}{x^2-1}$$ When one solves it: $$y = A(x^2-1)$$ Then we obtain an equation for a family of parabolas all intersecting $(-1,0)$ and $(...
1
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3answers
32 views

PARABOLA : Problem

Find the equation of line touching both the parabolas $$ x^2=-32y.......(1)$$ $$ y^2=4x.........(2) $$ i have equated slopes of both the parabolas and applied the condition that all the points on ...
2
votes
2answers
39 views

Tangents to a parabola from a point

Problem The angle between the tangent lines from the point $A(0,-1)$ to parabola defined as $y=x^2-ax+3$ is $135^{\circ}$. Then what could be the value of $a$? My attempt First I found that if $tan(...
6
votes
4answers
119 views

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 ...
3
votes
0answers
31 views

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 $D$. ...
0
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0answers
21 views

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 ...
0
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0answers
16 views

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 ...
0
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2answers
20 views

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 ...
-1
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1answer
34 views

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 ...
2
votes
2answers
44 views

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 ...
0
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1answer
37 views

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. ...
0
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3answers
48 views

Prove to be hyperbolae $x^2 - y^2 = a$ and $xy = b$ intersect at right angle.

Prove to be hyperbolae $x^2 - y^2 = a$ and $xy = b$ intersect at right angle. My idea: $$h_1:=x^2 - y^2 = a$$ $$h_2:=xy = b$$ By using implicit differentiation we can find $h_1'$ and $h_2'$. $$x^2 - ...
2
votes
3answers
33 views

$P$ is a point on ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ $(a>b)$ and $S$ and $S'$ are its focii

If $P$ is a point on ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ $(a>b)$ and $S$ and $S'$ are its focii. $\angle PSS'=\alpha$ and $\angle PS'S=\beta$, then prove that: $$ \tan\left(\frac{\alpha}{...
3
votes
2answers
54 views

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 ...
0
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1answer
29 views

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 ...
0
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1answer
20 views

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,...
0
votes
1answer
37 views

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 ...
1
vote
2answers
29 views

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 &...
0
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0answers
37 views

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 ...
0
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0answers
14 views

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 ...
0
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2answers
30 views

Parabola equation from cartesian to polar representation

I've got the following equation: 0) $ \frac{(y-y_p)^{2}}{4\cdot(x-x_p)} = p $ I'd like now to convert this expression to a polar representation. For this I got back to the basic rules: 1) $x = r\...
1
vote
2answers
23 views

Obtaining the equation of an ellipse with only information about the diameter and an angle

I am dealing with the following word problem: A spotlight throws a beam of light that is 25cm in diameter. If the beam hits the stage floor at an angle of $60 ^\circ$ with the horizontal, find an ...
1
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1answer
31 views

The slope of the tangent which touches both the parabolas $y^2 = 4ax$ and the parabola $ x^2=-32y$

The slope of the tangent which touches both the parabolas $y^2$ = $4ax$ and the parabola $x^2=-32y$ how do we find the slope of common tangent if I assume the slope of one of the cords and I find ...
1
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2answers
49 views

Range of $\alpha$ If tangents are drawn from external point to the Hyperbola

Two tangents can be drawn to the different branches of the hyperbola $$\frac{x^2}{1}-\frac{y^2}{4}=1$$ from the point $(\alpha,\alpha^2)$. Then Range of $\alpha$ is $\bf{My\; Try::}$If Line $y=mx+c$ ...
0
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0answers
22 views

Intersection between a hyper-paraboloid of revolution and a hyper-plane

I have the equation of a hyper-Paraboloid of revolution: $2cw=x^2+y^2+z^2$ and the equation of a hyperplane: $Ax+By+Cz+Dw+E=0$ These do intersect by my construction. How do I find the surface ...
0
votes
2answers
50 views

find the length of side

Tangents drawn to the parabola y2=4ax at the points P and Q intersect at T. If triangle TPQ is equilateral, then find the side length of this triangle. APPROACH P (at12 ,2at1) ; Q(at22 ,2at2) ; T (...
3
votes
1answer
41 views

Show that the equation of the normal line with the minimum y-coordinate is $ y = \frac{-\sqrt{2}}{2}x + {1\over k}$

Question: The curve in the figure is the parabola $y=kx^2$ where $k>0$. Several normal lines to this parabola are also shown. Consider the points in the first quadrant from which the ...
0
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2answers
38 views

How to prove that there's a plane with the required property?

I'm finding this particularly difficult. Let's say a circular cone is given with its base on a plane $\pi$. Then, if we cut this cone with planes that are not parallel to $pi$ we will have an Ellipse (...
0
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1answer
23 views

how to proceed in the following Question

Question What is the Locus of the foot of the perpendicular drawn from the centre of the ellipse $x^2$ +$3y^2$ = 6 I proceeded by assuming a pt (h,k) as the foot of the perpendicular to the tangent ...
0
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1answer
34 views

Finding the Locus of Circumcentre

Let $P$ be a point on circumcircle of $\Delta ABC$, where $A=(3,4), B=(-3,4), C=(4,3)$. Let feet of perpendicular from $P$ to $AB$ and $AC$ be $Q$ and $R$, respectively. Then locus of circumcentre of $...
1
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1answer
24 views

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 $x-axis$?
0
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1answer
41 views

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 ...
0
votes
1answer
93 views

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 ...
3
votes
4answers
79 views

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 ...
1
vote
3answers
65 views

The graph of the equation $x+y=x^3+y^3$ is the union of

The graph of the equation $x+y=x^3+y^3$ is the union of $(A)$line and an ellipse$(B)$line and a parabola$(C)$line and hyperbola$(D)$line and a point I tried to factorize the given equation. $x^3-x+...
2
votes
1answer
31 views

Finding the point on the ellipse under certain conditions

This is a kind of simple question, but it gives me hard time: An ellipse is given in coordinate system. It passes points $(a, 0)$, $(0, b)$, $(-a, 0)$, $(0, -b)$, where $a$ and $b$ are positive ...
0
votes
2answers
27 views

Incorrect Orientation of Graph in Stewart's Calculus 8E

The problem is #27 (matching graph), and the answer is VIII. I am losing my mind trying to figure out why graph VIII is oriented to have a greater set of Z vertices when X should be the major axis for ...
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0answers
27 views

Learn tracing of conics and concoids

A major portion of my course revolves around tracing of conics and concoids. But the explanation in my books is poor. I'm looking for some online notes/texts or videos to learn tracing of curves. I ...
1
vote
2answers
63 views

If the tangent at the point $P$ of an ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ meets the major axis and minor axis at $T$ and $t$ respectively

If the tangent at the point $P$ of an ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ meets the major axis and minor axis at $T$ and $t$ respectively and $CY$ is perpendicular on the tangent from the ...
1
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0answers
16 views

eccentricity of the conic

I'm given this question to find the eccentricity of this conic : $x^2 + ky = 0, k>0$ The given equation can be written as $x^2 = -ky$ now we can say compare this with the equation of parabola. But ...