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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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 = ...
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2answers
20 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
21 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
39 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 ...
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0answers
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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 ...
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2answers
47 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 ...
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1answer
37 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 ...
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30 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 ...
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1answer
18 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 ...
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1answer
26 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 ...
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19 views

Graphing on Desmos

I am trying to graph this picture of Smurfette on Desmos but am stuck.I cannot seem to figure out the equations needed / what equations to use. enter image description here
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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 $x-axis$?
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1answer
38 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 ...
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1answer
90 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 ...
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4answers
78 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 ...
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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. ...
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1answer
29 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 ...
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2answers
23 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
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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 ...
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2answers
62 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 ...
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0answers
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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 ...
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2answers
76 views

The graphs of $x^2+y^2+6x-24y+72=0$ and $x^2-y^2+6x+16y-46=0$ intersect at four points.Find the sum of the distances of these four points

The graphs of $x^2+y^2+6x-24y+72=0$ and $x^2-y^2+6x+16y-46=0$ intersect at four points.Compute the sum of the distances of these four points from the point $(-3,2).$ $x^2+y^2+6x-24y+72=0$ is a ...
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1answer
34 views

Rectangle $ABCD$ has area 200.An ellipse with area $200\pi$ passes through $A$ and $C$ and has foci at $B$ and $D$.

Rectangle $ABCD$ has area 200.An ellipse with area $200\pi$ passes through $A$ and $C$ and has foci at $B$ and $D$.Find the perimeter of the rectangle. Let the side lengths of the rectangle ABCD be ...
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1answer
42 views

An ellipse $x^2+4y^2=4$ is rotated anticlockwise through a right angle in its own plane about its center

An ellipse $x^2+4y^2=4$ is rotated anticlockwise through a right angle in its own plane about its center.If the locus of the point of intersection of a tangent to ellipse in its original position with ...
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2answers
62 views

If two points $P$ and $Q$ on the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ whose center is $C$ be such that $CP$ is perpendicular to $CQ$

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

Why are the contours of a cone equally spaced?

Having trouble understanding what makes the contours of a cone equally spaced, where f(x,y) = root(x^2+y^2). It would look something like this: ...
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2answers
65 views

If a chord joining the points $P(a\sec\alpha,a\tan\alpha)$ and $Q(a\sec\beta,a\tan\beta)$ on the hyperbola $x^2-y^2=a^2$ is a normal to it at $P$,then

If a chord joining the points $P(a\sec\alpha,a\tan\alpha)$ and $Q(a\sec\beta,a\tan\beta)$ on the hyperbola $x^2-y^2=a^2$ is a normal to it at $P$,then show that $\tan ...
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2answers
85 views

Prove that the value of $(abc)-(ab+bc+ca)+3(a+b+c)$ is $0$

If the points $\big(\frac{a^3}{a-1}, \frac{a^2-3}{a-1}),(\frac{b^3}{b-1}, \frac{b^2-3}{b-1}) ,\big(\frac{c^3}{c-1}, \frac{c^2-3}{c-1}\big)$ are collinear for three distinct values of $a,b,c$ and ...
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0answers
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Boundary curve of a surface (without Stokes)

So I've got to find the parametric equations of essentially a pringles chip. I've came up with a graph $$z = \dfrac{x^2}{11} - \dfrac{y^2}{28}$$ I've just got to figure out how to taper the graph now. ...
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1answer
55 views

Why does a parabola have a single point at infinity?

Consider the parabola $V(zy-x^2) \subset \mathbb P^2$. This parabola has only one point at infinity which is $[0:1:0]$. On the other hand we sketch the parabola, we see that there are two asympotes, ...
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1answer
43 views

Show that the arc length of sin(x) = circumference of ellipse

I've been staring at this problem for a while, and I would really appreciate any insight into finding the solution. The problem in its original state is as follows: Show that the arc length of the ...
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0answers
17 views

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 perpendicular to the focus and parallel to ...
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1answer
42 views

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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1answer
26 views

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 ...
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3answers
105 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 ...
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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
50 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
27 views

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 ...
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4answers
275 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 ...
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1answer
27 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 ...
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0answers
25 views

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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2answers
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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
30 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
65 views

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
28 views

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 ...
7
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2answers
132 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 ...
3
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1answer
32 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, ...
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0answers
16 views

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 ...