1
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1answer
26 views

Properties of the positive definite Hessian matrix of a convex function

I'm reading about nonlinear programming and I'm having trouble understanding the cool properties that a positive definite Hessian matrix $Q$ of $n$-dimensional function $f: \mathbf{R}^n\rightarrow ...
3
votes
2answers
53 views

How to determine the point at a set length along a given function (parabola)?

Given a specific function, a parabola in this instance, I can calculate the length of a segment using integrals to sum infinite right angled triangles hypotenuse lengths. My question is, can I reverse ...
0
votes
1answer
22 views

how to find the foci, directrix, center of a polar conic section. ($r=\frac{4}{5-4sin\theta} $)

I've been trying to figure this out for a bit and haven't found an answer. the equation is this: $r=\frac{4}{5-4sin\theta} $ I know I need to match this up to a conic graph so I divide top and ...
0
votes
3answers
50 views

Practise Calc Question

How is the radius $10$ in the circle equation: $x^2+y^2+6x-4y+3=0$? My work: Standard Form for circle: $(x-a)^2+(y-b)^2=r^2$ $X:$ $X^2+6x+[?]=-3+[?]$ $(x+3)^2= X^2+6x+9=-3+9 X^2+6x+9=6$ $Y:$ ...
1
vote
1answer
85 views

Finding the volume of a cone by integration of parabolic conic sections

I am working on a purely academic way of finding the volume of a right circular cone of height $h$ and radius $r$, (assume $h > r$), using integration of parabolic conic sections (conic sections ...
1
vote
2answers
125 views

Find the length of the longest line segment contained in the given region

Consider the region represented by the following in the $x-y$ plane. $y=v$, $x=u+v$ and $u^2+v^2\leq1$ $u$ and $v$ are parameters. What is the length of the longest segment contained in the given ...
1
vote
0answers
39 views

Finding the distance from a parabola (ballistic trajectory) to a point (for use in collision detection)

I need to have some form of collision detection / prevention for an object moving along a ballistic trajectory and a second stationary object on the same plane plane. The ballistic trajectory is ...
0
votes
1answer
64 views

Find the arc-length of the circle with radius a?

Find the arc-length of a circle with radius a. From the equation of a circle, I found out the equation for the one quadrant, which is: $y = \sqrt{a^2 - x^2}$ I tried solving the problem, and here's ...
-3
votes
2answers
147 views

Finding the maximum and minimum values on ellipse [closed]

Find the maximum and minimum values of f(x, y) = 5x + y on the ellipse x^{2} + 4y^{2} = 1
1
vote
1answer
37 views

Problem in conics question

A vertical line passing through the point ($h$,0) intersects the ellipse $$\frac{x^2}{4}+\frac{y^2}{3}=1$$ at the points P & Q.Let the tangents to ellipse at P & Q meet at the point R.If ...
0
votes
1answer
87 views

Proving properties of an ellipse

I'm studying about ellipse and its properties. My reference is the following pdf: http://nebula.deanza.edu/~bloom/math43/ellipse-derivation.pdf My questions are from the very first page of the ...
0
votes
4answers
218 views

Finding the maximum value of a function on an ellipse

Let $x$ and $y$ be real numbers such that $x^2 + 9 y^2-4 x+6 y+4=0$. Find the maximum value of $\displaystyle \frac{4x-9y}{2}$. My solution: the given function represents an ellipse. Rewriting it, we ...
2
votes
2answers
56 views

How to find equation of a tangent

How to find equation of a tangent on a $4x^2+9y^2-24x+18y+9=0$ in $T(6,-1)$? The solution is $x=6$, but I always get: $y = \frac{-4}{15}x + \frac {3}{5} $(?!) Alternate form: This is the ellipse ...
1
vote
1answer
149 views

Volume of ellipsoid bounded by two planes.

I need to find the volume of ellipsoid: $$5x^2 + {y^2\over25} + {3z^2\over4} = 1$$ if the ellipsoid is bounded by $z={-1\over2}$ and $z=1$ planes. I was able to find the total volume of the ellipsoid ...
1
vote
1answer
221 views

Parabola & Area Proving (Integral)

This is not a homework question. I am a new teacher (just graduated) and a student asked me this question. The points A(3,9) and B(-2,4) lie on the parabola y=x^2. The line y=x+6 joins A and B. The ...
1
vote
1answer
112 views

How to find the points of tangency of a parabola using Calculus?

How can someone find the points of tangency of a parabola in this situation? I need to find two points of tangency so that the triangle formed by the two tangent lines at those points and the x axis ...
0
votes
1answer
195 views

Implicit derivitave of a general ellipse

Consider an ellipse centered at the point $(h,k)$. Find all points $P=(x,y)$ on the ellipse for which the tangent line at $P$ is perpendicular to the line through $P$ and $(h,k)$. I know the general ...
0
votes
2answers
240 views

Intersection of two tangents on a parabola proof

There are two tangent lines on a parabola $x^2$. The $x$ values of where the tangent lines intersect with the parabola are $a$ and $b$ respectively. The point where the two tangent lines intersect has ...
0
votes
0answers
27 views

Determine the equation of the directrix of the parabola 2y=(x-1)(x-3) and find the equations of the tangents to the curve ..

Determine the equation of the directrix of the parabola 2y=(x-1)(x-3) and find the equations of the tangents to the curve at the points where the parabola cuts the x-axis Can someone please help me ...
1
vote
1answer
90 views

Parallel Curves to a Parabola

I have been modelling parallel curves to a parabola and realise if the parallel curve to a parabola is offset enough then the curve will overlap. I came across this research paper to explain why a ...
2
votes
2answers
175 views

Find parametrizations for Circles and Ellipses

a) The portion of the circle $x^2 + y^2 = 4$ traversed clockwise from $(-2,0)$ to $(0,2)$ b) The part of the ellipse $(x^2)/(4) + (y^2)/(9) = 1$ that lies above the line $y = 0$, traversed clockwise. ...
1
vote
0answers
75 views

Conic Sections - Points on a cone

On page 80 of Spivak's Calculus, 4th Edition, he writes: One of the simplest subsets of this three-dimensional space is the (infinite) cone illustrated in Figure 2; this cone may be produced by ...
2
votes
2answers
61 views

Finding the slope at a point $P(x_1,y_1)$ on a parabola

Given a point $P(x_1,y_1)$ on the graph of a parabola $y^2=4px$, prove that the slope at point P is $$\frac{y_1}{2x_1}$$
3
votes
1answer
45 views

How can I transform this equation in a conical?

In this equation $$2x²+y²-4x-6y+11=0$$ I got the result $(1,3)$ completing squares $2(x - 1)² + (y - 3)² = 0$   But on my list exercises, demanded that determine the foci, straight guideline ...
0
votes
1answer
231 views

Trajectory of a projectile.

From the definition of a parabola can we prove that the trajectory of a projectile is parabolic? And can this be proved by calculus?
1
vote
2answers
3k views

How do you find the distance between two points on a parabola

So I've been wanting to figure out a formula for an odd pattern I found... but to write a proof, I need to know one thing... How do I find the distance between two points on a parabola? Like, if I ...
1
vote
1answer
157 views

Car parking problem

I want to park my car doing similar to the one in the image. But I want to define a curve such that I park the car at once (without going forward, always backward). Suppose that the place that I want ...
2
votes
1answer
538 views

Minimum distance between $x = -y^2$ and $(0,-3)$

Find the minimum distance from the parabola $x + y^2 = 0$ (i.e. $x = -y^2$) to the point $(0,-3)$. This is a homework question. When I try to use the derivative and substitute $-y^2$ for $x$, I ...
1
vote
0answers
43 views

How can I find $\int{\sqrt{\left(b^2-1\right)x^2+1\over-x^2+1}}dx$?

I got this from the perimeter of an ellipse. I came up with the formula: arclength of f(x) for x from a to b=$\int_a^b\sqrt{f'(x)^2+1}dx$. Since an ellipse has the equation: $$\left({x-h\over ...
7
votes
4answers
148 views

closest point to on $y=1/x$ to a given point

I feel like I'm missing something basic - given a point $(a,b)$ how do I find the closest point to it on the curve $y=1/x$? I tried the direct approach of pluggin in $y=1/x$ into the distance formula ...
4
votes
4answers
446 views

Getting the equation of an ellipse using the constant and the foci

Find the equation of the ellipse with the foci at (0,3) and (0, -3) for which the constant referred to in the definition is $6\sqrt{3}$ So I'm quite confused with this one, I know the answer is ...
2
votes
0answers
133 views

My solution is right and the book is wrong (parabolas) or did I misunderstand it?

Find the equation of the parabola with the vertex at the origin; directrix 2x = 3 So what I did is, find the equation of the directrix $$x = \frac{3}{2}$$ and then because its the directrix, the ...
4
votes
3answers
237 views

What are the A,B,C parameters of this ellipse formula?

I am looking at $$A(x − h)^2 + B(x − h)(y − k) + C(y − k)^2 = 1$$ This is a rotating ellipse formula, where $h,k$ are the centroid of the ellipse. I have tried looking around for $A,B,C$ ...
1
vote
2answers
166 views

Derivation Of A General Equation Of An Ellipse

I am currently reading the topic alluded to in the title of this thread. In my textbook, after the equation has been derived, $\Large\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$, it says by finding the ...
1
vote
3answers
3k views

Length of a Parabolic Curve

I just wanted to know how I can find the length of a curve given by $f(x) = x^2$ from $x=0$ to $x=1$. For appproximation, the length is a bit larger than the hypotenuse of isosceles right triangle ...
3
votes
2answers
2k views

Possibly flawed calculus homework question (tangent to ellipse)

I have an online homework program called Web Assign for my calculus course. It has given me this problem: Find equations of both the tangent lines to the ellipse x^2 + 9y^2 = 81 that pass through the ...
1
vote
0answers
220 views

(Calculus 3) Having trouble finding the polar equation of a hyperbola.

Eccentricity e=sqrt(2), and one vertex is located at (2,0). I do know that if the vertex is located at (2,0), then the directrix is 2 units from the vertex. I am not sure how to find the location of ...
2
votes
2answers
2k views

general equation of a tangent line to a hyperbola

Suppose that there is a hyperbola of the form $\frac{x^2}{a^2}-\frac{y^2}{b^2} = 1$. I would like to figure out an equation that describes tangent line to this hyperbola. How would I be able to do ...
1
vote
0answers
109 views

History of calculus-based optimization

I would like to know: - who started with calculus-based optimization problems and when it was, - if there is a book focusing on history of ellipses/ conic sections - if someone ever tried to ...
2
votes
1answer
158 views

Finding vertex of a $f(x,y) = 0$ parabola

A parabola whose axis is oblique to the orthogonal coordinate axes is of the form $f(x,y)= 0$, for example $$f(x,y) = 9x^2 + 24 xy + 16 y^2 + 22x + 46 y + 9=0.$$ Using algebra only it is airly ...
0
votes
2answers
185 views

Maximum area of 3 rectangles inside an ellipse

I'm trying to determine the maximum area in a specific ellipse that can be filled with any 3 (horizontally aligned) rectangles. $$Ellipse: \frac{x^2}{36}+\frac{y^2}{16}=1$$ $$Area: ...
3
votes
1answer
159 views

How to check if two 2nd degree conic curves intersect in a given region?

Let there be two 2nd degree curves: $$f(x,y)=ax^2+by^2+cx+dy+e=0$$ and $$g(x,y)=fx^2+gy^2+hx+iy+j=0,$$ how is it possible to determine if these two curves intersect in some region, say $x \le 1 , y ...
2
votes
2answers
181 views

Finding minimum of the modulus of a 2 variable function

How to find the minimum value of $$|f(x,y)|$$ where $f(x,y)$ is a 2nd degree function in x and y with no 'xy' term. $$f(x,y)=ax^2+by^2+cx+dy+e$$ How is the process different from finding the minimum ...
3
votes
1answer
332 views

Finding minimum of a two variable 2nd degree function under a certain constraint?

How to find the the minimum non-negative value of a function: $$f(x,y)=ax^2+by^2+cx+dy+e$$ s.t. $x$ lies in $[0, A]$ and $y$ lies in $[A, \infty),$ where $A$ is a known constant. or simply $0\leq ...
1
vote
1answer
385 views

Calculate perimeter from parametric form with an ellipse?

Suppose I have a thing such as an ellipse: $$\left(\frac{x}{a}\right)^{2}+\left(\frac{y}{b}\right)^{2}=1$$ now we can define it so that $\frac{x}{a}=cos(\theta)$ and $\frac{y}{b}=sin(\theta)$. I ...
4
votes
2answers
306 views

Maximize the distance between a line normal to an ellipse and its center

My friend sent me this problem, which (upon Googling) seems to be from a Cornell class (1220?). Anywho. My advice to him was to parametrize the ellipse (say, in the first quadrant) with $x = a ...
0
votes
2answers
289 views

why we only have a approximation for every circumference for ellipse but not define a special formula for each ellipse

Why do we only have an approximation for every circumference for ellipse, but we cannot define a special ratio formula for each ellipse? Is it possible for people to use a computer to find the exact ...
1
vote
1answer
270 views

How to find points of tangency on a hyperbola?

If tangent lines to the hyperbola $9x^2-y^2=36 \;$ intersect y-axis at point $(0,6)$, find the points of tangency.
2
votes
4answers
3k views

How to find points of tangency on an ellipse?

The problem I have to solve is: If tangent lines to ellipse $9x^2+4y^2=36$ intersect the y-axis at point $(0,6)$, find the points of tangency.
5
votes
1answer
1k views

Formula for curve parallel to a parabola

I have a simple parabola in the form $y = a + bx^2$. I would like to find the formula for a curve which is parallel to this curve by distance $c$. By parallel I mean that there is an equal distance ...