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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1answer
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Focus of a parabola

If (2,0) is the vertex and y-axis the directrix of a parabola find the focus of the parabola. What does y-axis is directrix mean here?
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27 views

Partial differentiation and tangency

$\text{Q}$ Write the equation of tangent at the vertex of the parabola $2y^2+3y+4x-3=0$ . $\text{My attempt}$ if I partially differentiate the curve with y then I get $y+0.75=0$ which is correct ...
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2answers
48 views

Show that the curve $\dfrac{x^2}{a^2} +\dfrac{ y^2}{b^2} = 1$ form an ellipse

If the definition of an ellipse is the set of points $(x,y)$ such that given two focus points $F_1, F_2$ the sum of the distances from $(x,y)$ to each focus point is constant, how can one show that ...
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1answer
27 views

How to proove that foot of perpendicular drawn from focus to any tangent of an ellipse lie on auxillary circle?

One way is to find the foot of perpendicular and directly putting it into the equation of auxiliary circle. But that is quite a lengthy proof, is there any other short method to prove this property?
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1answer
60 views

What practical purpose — or application — do directrices serve?

In Calculus II (and briefly in Trigonometry, if I remember correctly) the concept of a directrix began poking its head around conic sections. While covering parabolas, ellipses, and hyperbolas, the ...
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2answers
38 views

Can a line be a taxicab parabola?

For example, a line segment can be a taxicab ellipse if the sum of the distances equals the distance between the foci. So, can a line be a taxicab parabola?
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2answers
112 views

Locus of a point on a fixed-length segment whose endpoints slide along orthogonal lines

Suppose we have some segment $AB$ of constant length that slides in such a way that its endpoints are moving along orthogonal lines. Let $P$ be a point in the segment so that $|AP| = a$ and $|PB| = b$....
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1answer
26 views

Can a line parallel to axis of parabola also represent tangent at a point along with the one whose slope is found using calculus?

Consider a parabola with the equation $y^2=4x$ its axis is the x-axis and vertex is (0,0) and focus at (1,0). Consider any point on the parabola say (4,4). Now we define tangent at this point as a ...
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0answers
38 views

HYPERBOLA : Problem [duplicate]

If two points $P$ and $Q$ on the hyperbola $\frac{x^2}{a^2} -\frac{y^2}{b^2} = 1$ whose centre is $C(0,0)$ are such that $CP$ is perpendicular to $CQ$ , $a<b$ , then prove that $$\frac{1}{(CP)^2} ...
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1answer
19 views

equation of an ellipse given its center and two tangent lines

There exists an ellipse centered at (0,0) with two tangent lines given by $y=-\frac{1}{2}x + \frac{\sqrt{39}}{2}$ and $y=\frac{1}{3}x + \frac{7}{3}$. Find the ellipse. So, I used the equation of one ...
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1answer
26 views

Eccentricity of a hyperbola given the angle between the x-axis and its asymptote

I need to find the eccentricity of a hyperbola whose asymptote makes an angle $\alpha$ with the $x$-axis. So, I take the case where the transverse axis will be horizontal, $i.e.$ $\frac{x^{2}}{a^{2}}...
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1answer
40 views

Ellipse and chord length

There is a analytic geometry problem: In the ellipse $\frac{x^2}{4}+y^2=1$, segment $AB$ is a chord and $AB=\sqrt{3}$, find the maximum and minimum area of $\triangle AOB$. My progress Assume ...
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0answers
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Program for General Form

I am trying to find the general form of the conic sections, however, I am not able to create a program that runs it. I know the way you find it is by using B^2-4AC but I would like to write a program ...
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8answers
421 views

How do I transpose an ellipse function to stretch the ellipse into curved space?

I'm working on an engineering project, using CAD software. I can write simple parametric functions to draw an ellipse, with $\theta$ ranging from $0$ to $2\pi$ radians:$$x=3.75\cos\theta$$$$y=1.25\...
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1answer
56 views

Ellipse's farthest point to another point

I am trying to find the farthest and closest points of a ellipse without using any brute force type of coding. The processing power is limited so it should be as pinpoint as possible. I have tried a <...
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0answers
9 views

Calculating X & Y coordinates of a point that is perpendicular to an ellipse point AND offset by -5

I am trying to calculate an offset point from a point on an ellipse - I need to be perpendicular to each point on the ellipse but 5 points in from the point on the ellipse. The result will probably ...
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4answers
223 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 ...
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0answers
27 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 ...
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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 $\frac{xx_0}{a^2} - \frac{yy_0}{b^2} = 1$ ...
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4answers
70 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 ...
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2answers
68 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 ...
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1answer
37 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
27 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: $$\lim _{ n\rightarrow { 0 }^{ + } }{ \frac { \cosh { (nx) } -1 }{ \cosh { n } -1 } } = { x }^{ 2 }$$ ...
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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 ...
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1answer
68 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 ...
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1answer
35 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 ...
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0answers
30 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 $(...
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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 ...
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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(...
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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
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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 $D$. ...
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0answers
20 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 ...
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0answers
12 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 ...
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2answers
18 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 ...
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0answers
25 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
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2answers
42 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 ...
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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. ...
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3answers
46 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
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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}{...
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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 ...
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1answer
28 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 ...
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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,...
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1answer
34 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 ...
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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 &...
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0answers
36 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 ...
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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 ...
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2answers
24 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\...
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2answers
22 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 ...
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1answer
25 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 ...
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2answers
45 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 $y=mx+c$ ...