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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Prove that the equations of common tangents to the two hyperbolas $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ and $\frac{y^2}{a^2}-\frac{x^2}{b^2}=1$ are

Prove that the equations of common tangents to the two hyperbolas $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ and $\frac{y^2}{a^2}-\frac{x^2}{b^2}=1$ are ...
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1answer
14 views

Volume between hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ and line $x = 2a$ around $y$ axis

I'm trying to calculate the volume between the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ and the line $x = 2a$ around the $y$ axis using two methods but I'm getting different answers: Using ...
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3answers
31 views

Confusion regarding slope of a tangent to a parabola

I had learnt that differentiating the function $y=f(x)$ and putting the value of a point $(x_1,y_1)$ would give the slope of the tangent to the function at $(x_1,y_1)$. In other words, to find the ...
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1answer
43 views

Does an ellipse or circle have greater circumference?

If an ellipse has semi-major axis length a, and a circle has radius a, and you walked along their boundary, which one would be longer? A circle's circumference is calculated using $2\pi r$, but I ...
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1answer
21 views

Tangents are drawn from the point $(\alpha,\beta)$ to the hyperbola $3x^2-2y^2=6$ and are inclined at angles $\theta$ and $\phi$ to the $x-$axis.

Tangents are drawn from the point $(\alpha,\beta)$ to the hyperbola $3x^2-2y^2=6$ and are inclined at angles $\theta$ and $\phi$ to the $x-$axis.If $\tan\theta.\tan\phi=2,$ prove that ...
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0answers
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The difference of the focal semi axes of an ellipse and a hyperbola is equal to $4$.If the ratio of their eccentricities is $\frac{3}{7}$.

An ellipse and a hyperbola have their principal axes along the coordinate axes and have a common foci separated by a distance $2\sqrt{13}$,the difference of their focal semi axes is equal to $4$.If ...
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2answers
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A circle has the same center as an ellipse and passes through the foci $F_1$ and $F_2$ of the ellipse, two curves intersect in $4$ points.

A circle has the same center as an ellipse and passes through the foci $F_1$ and $F_2$ of the ellipse,such that the two curves intersect in $4$ points.Let $P$ be any one of their point of ...
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1answer
39 views

How to find the center of an Ellipse, given a focal point, radius, and eccentricity

I am attempting to create a small computer simulation with planetary orbits. Calculating the position a planet has on its orbit at a certain time works fine. However, I now want to draw their orbits ...
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1answer
44 views

Graphing of $y = (x^2 - a)^2$

I was graphing the equation $y = x^2 - a$ and I know why the graph is a parabola intersecting at the points $(-1,0)$ and $(1,0)$. However, when I graph $y = (x^2 - a)^2$, the graph oddly changes, as ...
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0answers
13 views

Chord of one ellipse tangent to other

After finding equation of PQ I tried putting value of y from PQ in other ellipse and then set discriminant=0. But it is getting too tedious
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2answers
70 views

How to prove parametric equation of a ellipse

The parametric equation of a ellipse is $$x=a \cos t\\y=b \sin t$$ It can be viewed as $x$ coordinate from circle with radius $a$, $y$ coordinate from circle with radius $b$. How to prove that it's ...
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2answers
58 views

Finding the locus of the points of intersection of tangents to a parabola

2 tangents to the parabola $y^2=4ax$ meet at an angle of $45^\circ$. Prove that the locus of their point of intersection is $y^2-4ax=(x+a)^2$. $$$$ I got completely stuck with this question. All ...
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1answer
23 views

Find the length of latus rectum of the conic $7x^2+12xy-2y^2-2x+4y-7=0$.

Find the length of latus rectum of the conic $7x^2+12xy-2y^2-2x+4y-7=0$. The given conic $7x^2+12xy-2y^2-2x+4y-7=0$ is a hyperbola because when i compare it with $ax^2+2hxy+by^2+2gx+2hy+c=0$ and ...
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0answers
22 views

Length of a focal chord [duplicate]

how to prove that that length of focal chord of standard ellipse(a>b) which inclined angle titha to the major axis is 2ab^2/(a^2sin^2θ+b^2cos^2θ I tried Equation of focal chord inclined at angle ...
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1answer
43 views

Find the eccentricity of the ellipse $(x-3)^2+(y-4)^2=\frac{y^2}{9}$

Find the eccentricity of the ellipse $(x-3)^2+(y-4)^2=\frac{y^2}{9}$ $(x-3)^2+(y-4)^2=\frac{y^2}{9}$ $x^2-6x+9+y^2-8y+16-\frac{y^2}{9}=0$ $(x-3)^2+\frac{8y^2}{9}-8y+16=0$ ...
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2answers
60 views

Locus of intersection of two lines

If the tangent at any point P of a circle $x^2 + y^2 = a^2$ meets the tangent at a fixed point A $(a,0)$ in T and T is joined to B , the other end of the diameter through A . Then we have to prove ...
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2answers
38 views

How to use parametric equation/trigonometric identity to show an ellipse?

I have the equation $16x^2+25y^2=400$, and the parametric equation $(x,y)=(5\cos t, 4\sin t)$. If I plug in the parametric equation into the first equation, I end up with the trigonometric identity ...
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2answers
54 views

Finding the number of normals to a parabola

Find the number of normals to the parabola $y^2=8x$ through (2,1) $$$$ I tried as follows: Any normal to the parabola will be of the form $$y=mx-am^3-2am$$ Since the point (2,1) lies on the ...
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1answer
25 views

Problem on finding the focus of parabola $4y^2+12x-20y+67=0$

Find the focus of the parabola $4y^2+12x-20y+67=0$ $$$$ I tried proceeding as follows: $$(2y-5)^2=-12x-42$$ $$(2y-5)^2=4(-3x-\frac{21}{2})$$ This is of the form $$Y^2=4aX$$ where $a=1, Y=2y-5, ...
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1answer
15 views

how to find hyperbola equation knowing tangent line and point

I have a problem. A hyperbola passes through point $(3,2)$ and $9x+2y-15=0$ is a tangent line. Find the equation of hyperbola.
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4answers
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Find equation of tangent line knowing hyperbola and point on line

I have a problem I've been trying to solve, but I was not able to do it. A hyperbola is $x^2-y^2=16$ and a point is $(-1,-7)$, not on the curve. Find equation of tangent line to the hyperbola ...
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1answer
34 views

Equation of parabola whose ends of latus rectum are $(-1,2)$ and $(5,2)$

I found the distance between ends using distance formula i.e $6$. $\Rightarrow 4a => $ $a= 3/2$ and the focus $(2,2) $ What should I do next? How to use this information in $(x-h)^2 = -4a(y-k)^2$
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4answers
68 views

Using polar coordinates to find the area of an ellipse

Considering an ellipse with the $x$ radius equal to $a$ and the $y$ radius equal to b$:$ I figured that some kind of parameterization might be: $x=a\cos\theta$ $y=b\sin\theta$ and then polar ...
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2answers
43 views

Normal lines to a parabola, and areas bounded by them

This is the question: What I have done: (a) Show that the equation of the normal to the parabola at a point $(x_0,y_0)$ is $y = {-1\over 2kx_0} + kx_0^2 + {1\over 2k}$ $$ f(x) = kx^2 $$ $$ ...
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1answer
34 views

Axis of symmetry of parabola

I have equation of parabola $(ax+by)^2+2fy=0$ and I have to find axis of this parabola so I made the substitution $X = ax +by$ and $Y= \frac{x}{a}-\frac{y}{b}$ and then solving by these substitution I ...
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2answers
35 views

A ball is thrown upward from the top of a tower.If its height is described by $-t ^2+60t+700$,what is the greatest height the ball attains?

A ball is thrown upward from the top of a tower.If its height is described by $-t ^2+60t+700$,what is the greatest height the ball attains ? I've managed to get a solution by realizing that ...
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1answer
43 views

Ellipse Area (Trouble understanding answer)

Question: An elipse with equation $$ {x^2\over a^2} + {y^2\over b^2} = 1 $$ is enclosed by the hyperbolas given by $xy=1$ and $xy=-1$. , Determine the largest area of an ellipse enclosed ...
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1answer
39 views

Locus of vertex of a rectangle

If from the vertex of a parabola $y^2 = 4ax$ a pair of chords be drawn at right angles to one another and with these chords as adjacent sides a rectangle be constructed , then we have to find the ...
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2answers
45 views

Find the tangent equation to the circle

The circle is given as $$x^2+y^2+z^2-7y+2z-8= 0$$ $$3x-2y+4z+3=0$$ at the point $(-3,5,4)$. I know the answer will be in the form of $$\frac{(x+3)}{l}=\frac{( y -5 )}{m}=\frac{( z-4)}{n}$$ but ...
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1answer
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3 normals on a parabola

If $(x_1, y_1), (x_2, y_2)$ and $(x_3, y_3)$ be three points on the parabola $y^2 = 4ax$ and the normals at these points meet in a point then how will we prove that $$ \frac{x_1 -x_2}{y_3} + ...
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1answer
70 views

Chord of a parabola $y^{2}= 4ax$

Prove that on the axis of any parabola $y^2=4ax$ there is a certain point $K$ which has the property that,if a chord $PQ$ of the parabola be drawn through it ,then $$\frac{1}{PK^2}+\frac{1}{QK^2}$$ is ...
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1answer
53 views

The point of intersection of two perpendicular tangent lines to a parabola

If two perpendicular straight lines through the focus of the parabola $y^2 = 4ax$ meet its directrix in $T $ and $T'$ respectively. Show that the tangents to the parabola parallel to the ...
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2answers
44 views

Define ellipse and hyperbola in terms of distances from a point

Just as we say a circle is a locus of points that are equidistant from a single point. How to define an ellipse and a hyperbola in a similar way?
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1answer
162 views

Geometric derivation of the quadratic equation

The quadratic equation can be thought of as specifying distances in the Euclidean plane. It tells us that the $x$-intercepts of a function occur at a distance of $\frac{\sqrt{b^2-4ac}}{2a}$ from the ...
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0answers
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Proof of equation of ellipse

The ellipse can be defined as a conic section with eccentricity lesser than unity. How can you derive the equation of the ellipse using this definition? I can't find proof of the formula using this ...
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0answers
22 views

Polar form of generalized superellipse

I am looking for the polar form of the generalized superellipse: $$ \left|\frac{x}{a}\right|^{n_2}+\left|\frac{y}{b}\right|^{n_3}=1 $$ where $a$ and $b$ are the semi major and semi-minor axes. I have ...
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2answers
28 views

Given Function, find domain and description of graph $y = f(x)$

I am studying for Graduate Record Exam. The following question is difficult. Given the domain and description of $f(x) = 5 - (x + 20)^2$, including its shape, and the $x$ and $y$-intercepts To find ...
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Construction new ellipse

Using a pencil the thread was pulled on the ellipse. Then the pencil started to rotate around the ellipse. How to prove that a new geometric figure which the pencil drew is also an ellipse (with the ...
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3answers
342 views

Need to find the ellipse of maximum area inscribed in a semicircle.

An ellipse inscribed in a fixed semi circle touches the semi-circular arc at two distinct points and also touches the bounding diameter. Its major axis is parallel to the bounding diameter. ...
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1answer
41 views

How would I put $x^2 + 4x + 25y^2 - 50y = -4$ into the equation for an ellipse?

The equation is $x^2 + 4x + 25y^2 - 50y = -4$. How would I put this into the equation for an ellipse $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$?
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3answers
17 views

Show that this section of the cone is a hyperbola

Show that section of a cone, with vertex at origin and base $x=a$ & $y^2+z^2=b^2$, intersected by a plane parallel to $XY$ axes is a hyperbola.
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3answers
58 views

What is the general equation of an ellipse that is not aligned with the axis?

I originally asked this in an answer to the following question: What is the equation of an ellipse that is not aligned with the axis?. As I noted in the opening paragraph I DO NOT HAVE THE NECESSARY ...
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1answer
57 views

Can't Simplify this equation for a Ellipse(Complex Numbers)

I'm asked to sketch the set $\{z \in C : |z + i| + |z + 1| = 2\}$. I've gotten to the point where I've got the modulus form of $|z + i| + |z + 1|$: $$\sqrt{x^2+(y+1)^2} + \sqrt{(x+1)^2+y^2} = 2$$ How ...
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1answer
19 views

Problem regarding parabola

While studying conic sections, in the parabola portion, I read that The sum of the ordinates of the extremities of the chords of the parabola $y^2=4.a.x$ which are parallel to each other is ...
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1answer
36 views

Parabola conic section

Two tangents to the parabola $y^2= 8x$ meet the tangent at its vertex in the points $P$ and $Q$. If $|PQ| = 4$, prove that the locus of the point of the intersection of the two tangents is $y^2 = 8 ...
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2answers
38 views

Finding area of rectangle under a parabola asymmetrical with respect to the Y-axis: What did I do wrong?

I am using these as references: How to find the dimensions of a rectangle if its area is to be a maximum? Does the symmetry of a parabola in finding the maximum area of a rectangle under said ...
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1answer
41 views

Does the symmetry of a parabola in finding the maximum area of a rectangle under said parabola matter?

Apologies for my English, I'm a not a native speaker... So I've got this homework, about finding the maximum area of a rectangle under a parabola. I'm using this as a reference to do my work: How ...
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2answers
61 views

Determine the dimension of the set of surfaces of $\mathbb{P}^{3}$ that contain certain conic.

Let $C\subseteq\mathbb{P}^{3}$ be the conic of equations $$ C=V(X_{3}, X_{0}X_{2}-X_{1}^{2})=\{(t_{0}^{2}:t_{0}t_{1}:t_{1}^{2}:0)\in\mathbb{P}^{3}:(t_{0}:t_{1})\in\mathbb{P}^{1}\}. $$ I have to ...
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1answer
18 views

orthogonal diagonalization to sketch equation: $5x^2-24xy-5x=13$

to sketch this i wrote the equation down in the form: $X^TAX=13$ where $X^T=[x\:\:y]$ and $A=\begin{bmatrix} 5&-12\\ -12&-5 \end{bmatrix} $. Then, $A$ is orthogonally diagonalizable so there ...
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0answers
37 views

Prove an ellipse is unique if the foci and a tangent are given.

Given 2 points $F_1$ and $F_2$ and a straight line $l$ which does not cross $[F_1F_2]$. Prove that there exists an unique ellips with $F_1, F_2$ as foci, and tangent $l$. What if $l$ crosses ...