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
32 views

Distances in HP

A variable straight line passes through the fixed point $A(6,1)$ and meets the ellipse $x^2 + 2y^2 = 2$ at points $B$ and $C$. If $P$ is a point such that the lenghts $AB, AP, AC$ are in HP (harmonic ...
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2answers
21 views

Axes Rotation Problem

Given $$x^2 - 4xy + 5(\sqrt5y) + 4y^2 + 1 = 0$$ rotate the axes to eliminate the $xy$-term in the equation, then write the equation is standard form.
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1answer
19 views

Hyperbola with its directrix

The equation $9x^2 - 16y^2 -18x +32y-151=0$ represents a hyperbola . We have to find the equation of its directrix. I simplified the equation and got : $$(3x-1)^2 -(4y-1)^2 = 151$$ And found that ...
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0answers
47 views

Finding a mirror point on a parabola

What is the height of the ball at a point of 3 metres beyond where it was thrown, measured horizontally? How far is the ball from where it was thrown when its height has this value again? ...
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19 views

Geometric Invariants of a conic section

There are three independent invariants for every conic section, viz., $$[I_1,I_2,I_3]= [ (a + b + c), (a b -h^2), Det(( a,h,g), (h,b.f), (g,f,c) )] $$ How are they related to the known geometric ...
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3answers
40 views

A circle pass through origin and centre is $(3,-3)$ and find coordinated point on the circle [closed]

A circle pass through origin and centre is $(3,-3)$ and line $y=x-6$ meet the circle at point $P$ and $Q$. Find coordinated of point on the circle where tangent are parallel to line $PQ$. I got the ...
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0answers
12 views

Directrix and foci relationship on a hyperbola

on a parabola the distance from the vertex is equal to the distance from the directrix. Is this the case with hyperbolas? I have looked on multiple math websites and they don't state this but from ...
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2answers
67 views

Find the eccentricity of a conic

Find the eccentricity $e$ of the conic $$S \equiv 39x^2+11y^2-96xy+14x+2y-34=0.$$ My try: Comparing with general second degree conic $$ax^2+2hxy+by^2+2gx+2fy+c=0$$ we have $a=39$, $b=11$, $2h=-96$, ...
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3answers
48 views

Minimum Enclosing Ellipsoid To Maximal Enclosed Ellipsoid

Given a convex body $K$ and an ellipsoid of minimal volume which contains $K$, find the maximal ellipsoid contained in $K$. I have tried to multiply the matrix by 4 (since the eigenvalues are the ...
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1answer
30 views

Help with a geometrical problem on ellipses

Let $E$ the ecllipse, with center $O = (0,0)$, focus $F=(4,0)$ and vertex $V=(5, 0)$. Let $N$ be a point on the ellipse $E$, and let $Q$ be the orthogonal projection of $N$ onto the y-axis. Find the ...
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1answer
60 views

Volume bounded between an Ellipsoid and a Cone?

I'm a bit confused about how I would be able to find the volume bounded by a cone of known theta and an oblate spheroid of b = c. I'm trying to use triple integrals for the solution, and I think I ...
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1answer
24 views

Geometrical place of a point R

Given the ellipse E $4x²+9y²=36$ and a point $P=(4,7)$. Let $Q=(x,y)$ point of the ellipse and $R$ a simetric point of Q respect $P$. Find the geometrical place of $R$. Ok, i think that R belong to a ...
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1answer
17 views

Number of common chords of the two parabolas

How can we find common chords to the parabolas $$ (y-2)=(x-3)^2$$ and $$(x-2)=(y-3)^2$$ without drawing graphs. What i have done is i have subtracted both of them and i got $$(y-x)=(x-y)(x+y-6)$$ ...
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2answers
30 views

Maximum area of $\Delta QSR$

The circle $C \equiv x^2+y^2=1$ cuts $X$ and $Y$ axes at $P$ and $Q$ Respectively. if another circle with centre $Q$ and variable radius is drawn so that it meets $C$ at $R$ and the line $PQ$ at $S$. ...
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1answer
47 views

Area of Triangle in ellipse

Full question: Prove that the area of the triangle formed by three points of an ellipse, whose eccentric angles are $\theta , \phi$ and $\psi$ , is ...
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1answer
29 views

Find the equation of the parabola with focus at $(-1,0)$ and vertex at $(3,0)$

For this problem I cannot figure out how to find the directix, defined to be perpendicular to the axis of symmetry. The general point $(x,y)$ on parabola needs to be far from $(-1,0)$ as it is from ...
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0answers
24 views

Hyperbola: A case of an ellipse?

Can i treat a hyperbola as a special case of ellipse. Like substituting b^2 with -b^2 Would all things still work? And also, why is a parabola different from the family of (circle, ellipse, ...
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0answers
12 views

Diagonal of parallelepiped circumscribed around ellipsoid is constant

There are many rectangular parallelepipeds that can be circumscribed around a given ellipsoid in $\mathbb R^n$. Prove that the length of the main diagonal does not depend on the choice of such ...
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3answers
59 views

Find equation of circle in the first quadrant touches $x$-axis $y$-axis and straight line $3x-4y-20=0$ . The point $H(12,4)$ lies on the straight line

1)Find equation of circle 2)Equation of another tangent from point $H$ to the circle The circle in the first quadrant touches $x$-axis $y$-axis and straight line $3x-4y-20=0$. The point $H(12,4)$ ...
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4answers
46 views

Polar equation of an ellipse given the origin coordinates and major and minor axis lengths?

I've been trying to create a polar equation that will give me all points on an ellipse with the independent variable being theta and the dependent variable being the radius, but I'm having a great ...
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1answer
43 views

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
20 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
34 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
57 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
30 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
25 views

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
78 views

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
113 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
45 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
20 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
100 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
70 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
27 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
50 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
85 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
57 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
73 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
17 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
39 views

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
55 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
76 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
59 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
42 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
40 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 ...
3
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0answers
52 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 ...
2
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1answer
52 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 ...
1
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1answer
18 views

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} + ...