# Tagged Questions

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### Recognize conics from the standard equation

Suppose $Ax^2+Bxy+Cy^2+Dx+Ey+k=0$ is a conic in the Euclidean plane. How do I recognize what is it? In my book they have proved the determinant test that if $B^2-4AC$ is $>0$ if hyperbola, $=0$ if ...
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### Ellipse, hyperbola and principle axis

Would anyone mind telling me how to solve (a)? I have no idea what I should do to solve this problem. Also, what is principal axes?
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### 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 ...
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### Ellipses given focus and two points

I would like to find all ellipses which contain 2 given points and has one focus at origin (zero). All in 2D plane. There are several possible approaches but I'm not sure which is the best - both ...
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### Why are two definitions of ellipses equivalent?

In classical geometry an ellipse is usually defined as the locus of points in the plane such that the distances from each point to the two foci have a given sum. When we speak of an ellipse ...
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### Locus Problem .

Prove that the locus of the middle points of all tangents drawn from points on the directrix to the parabola is $y^2(2x+a)=a(3x+a)^2$
I was working on this and I wanted to be sure I wasn't too far off. Given: $\frac{\alpha}{r} = 1 + \epsilon \cos \theta$ where $\epsilon$ is eccentricity. Also $\frac{(x + x_0)^2}{A^2} = ... 1answer 45 views ### Product of the distance from foci to a tangent is a constant I am supposed to determine what is the result of said product. Given$P(x_0,y_0)$, I need to calculate the distance from the foci to the tangent line that passes through$P$, and then multiply the ... 1answer 174 views ### Find locus of points relating to an ellipse I would like to find the equation of the following locus. For a big circle C centered at (0,0), the locus of points that the sum of distances to Y-axis and to C is 1, say in the first quadrant, is ... 1answer 67 views ### Common Normal Parabola Problem Prove that two parabolas$y^2=4ax $and$y^2=4c(x-b)$cannot have a common normal other than the axis, unless$ b/a-c>2$. I couldn't think of a satisfactory approach. Please Help. 2answers 58 views ### Proving a parabola property I need help with this question that i attempted to solve using the equation$y^2=4ax$: "Prove that on the axis of any parabola there is a certain point which has the property that,if a chord PQ of ... 1answer 286 views ### Ellipse in Cartesian and in Polar Coordinates So I was studying about ellipses in Polar Coordinates, and the book said Let F be a fixed point, and l be a fixed line in a plane. Let e be a fixed positive number. The set of all points P in ... 1answer 109 views ### Enlarging an ellipses along normal direction Given an ellipses, enlarge it along normal direction a fixed length say 1cm. Do we get another ellipses? If so, how to prove ? 2answers 133 views ### How many times can quadric kiss cosine at given point? Let a quadric$ax^2+2bxy+cy^2+dx+ey+f=0$touches the plot of$y=\cos(x)$at the point$(0,1)$with multiplicity$n$. What is the maximum possible value of$n$? Recall that a joint point$P$of ... 1answer 193 views ### Find equations of the ellipses given conditions on the directrices, foci, and vertices The ellipses have their centers at the origin and their major axes on the$x$-axis. Find the equation: with distance between directrices$27$, and between foci$3$; with a focus at$(-\sqrt{13},0)$... 1answer 57 views ### Finding a,b of elipse Given$x^{2}+y^{2}=R^{2}$, so that we multiply every$x$by$a$and every$y$by$b$,$(a>b)$And the distance between the focuses of this locus is$48R$, and the area of the rhombus which ... 0answers 46 views ### Equation of a general conic from 3 points and the major axis I have read that given 3 points on a conic and the equation ($ax+by+c=0$) of its major axis, we can write the equation of the conic ($Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$). I've seen it done by ... 0answers 92 views ### Parabolic segment problem I have a problem. I have tried to solve it but I get$125/6$instead of$9/2(textbook result). Find the area of the parts of plane given by the solutions of the following system: $$... 1answer 43 views ### How does this method to find the centre work? Say we have a conic with equation f(x,y)=c. My teacher says that it's centre satisfies the equations : f_x(x,y)=f_y(x,y)=0 (If it has a centre). She didn't give any explanation. I thought this ... 2answers 101 views ### Prove that |PF_{1}|+|PF_{2}| is Constant in an Elipse Given an elipse with two focus F_{1} an F_{2}, and A is an arbitrary point at the elipse. Stright line AF_{1} has another intersection point B with the elipse, and AF_{2} has another ... 1answer 92 views ### In an equation that looks like the standard form of an ellipse, what must the constant on the RHS equal for exactly one solution? I am working on a homework question: What must be the value(s) of c for the following equation to have exactly 1 solution? The equation is of the standard form of the equation for an ellipse, ... 4answers 137 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 ... 2answers 354 views ### Find the standard form of the conic section x^2-3x+4xy+y^2+21y-15=0 Find the standard form of the conic section x^2-3x+4xy+y^2+21y-15=0. I understand the approach in trying to solve these problems. But the 4xy is confusing me. I am not sure of where to start on ... 2answers 457 views ### Foci of Ellipse - given: Width and Height Can you help me out with the next problem. I have an ellipse based on a width and a height. Is there any way you can find out where the focal points are? I need this information because I need to ... 3answers 270 views ### The equation of an ellipse I have a couple of questions regarding ellipses. Get the equation of the ellips With Foci (\pm 3,0) and which goes through (2,\sqrt{2}). This one I didn't understand AT ALL. I need some ... 1answer 136 views ### Equation of a parabola: Translations and directrixes Find the equation of the paraboles, with: Focus (3,0) and x=-3 is the directrix Focus (0,2) and y=-2 is the directrix Vertex (I believe it is the vertex, the lowest/highest point) (1,2) ... 1answer 244 views ### How to find intersection of an ellipse and a line that passes through the foci There are two lines, parallel to the x-axis, which pass through the foci and intersect the ellipse at four points. How can I find the points of intersection? vertex: (0,0) foci: (0,10) and ... 1answer 217 views ### Prove that a conic section is symmetrical with respect to its principal axis. A Calculus book that I'm self-studying is asking me to prove the following theorem about conic sections: A conic section is symmetrical with respect to its principal axis. Here is my attempt at ... 2answers 163 views ### Find the parallels to a line which are tangent to an ellipse Having the equation of a line, how can I find which of its parallels are tangent to an ellipse of equation x^2 + 9y^2 = 1? If the equation of the line is y = mx + q, I know that its parallels ... 1answer 311 views ### What is the path equation that is created with the middle point of a fixed length line segment that touching both ends to an ellipse. Ellipse equation is (\frac{x}{a})^2+(\frac{y}{b})^2=1 and the length of line segment is 2k, if we move the line segment all around of the ellipse while touching both ends to the ellipse. What is ... 2answers 601 views ### Finding & Plotting equation of hyperbola given foci, and difference in distances between them. I have to plot the hyperbola (3 of them actually) in MATLAB, and so it'd be good if I could find some sort of general formula. The foci do not necessarily have to be on the axes (e.g. (5,3) and ... 2answers 240 views ### Conditions for intersection of parabolas? What are the conditions for the existence of real solutions for the following equations:$$\begin{align} x^2&=a\cdot y+b\\ y^2&=c\cdot x+d\end{align}$$where a,b,c,d are real numbers. ... 2answers 4k views ### How do I find the equation of a tangent line to a curve? I'm given x^2+2x-4 at x=2 and I have to find the tangent line to this curve at that point... 1answer 188 views ### Need help with the proof of conic section Prove that the intersection of a plane and a object consist of one cone and one upside-down cone where the tip of cone meet is either degenerate conic or conic Also, idenify in what situation, the ... 2answers 160 views ### Hyperbola property I am posting the following question under homework category. I hope I will have very good answer from mathematicians about conic sections. I have seen closely the conic sections and their ... 3answers 268 views ### Apostol Section 13.25 #13 - Conic Sections Question: Prove that a similarity transformation (replacing x by tx and y by ty) carries an ellipse with center at the origin into another ellipse with the same eccentricity. (The next ... 1answer 296 views ### Why do definitions of distinct conic sections produce a single equation? I understand how to get from the definitions of a hyperbola — as the set of all points on a plane such that the absolute value of the difference between the distances to two foci at (-c,0) and ... 3answers 580 views ### Parametric form of an ellipse given by ax^2 + by^2 + cxy = d If c = 0, the parametric form is obviously x = \sqrt{\frac{d}{a}} \cos(t), y = \sqrt{\frac{d}{b}} \sin(t). When c \neq 0 the sine and cosine should be phase shifted from each other. How do I ... 1answer 890 views ### Formula for curve parallel to a parabola I have a simple parabola in the form y = a + bx^2. I would like to find the formula for a curve which is parallel to this curve by distance c. By parallel I mean that there is an equal distance ... 3answers 894 views ### Canonical to Parametric, Ellipse Equation I've done some algebra tricks in this derivation and I'm not sure if it's okay to do those things.$$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\frac{x^2}{a^2} + \frac{y^2}{b^2} = \cos^2\theta + ... 3answers 191 views ### What is the most direct way to derive an equation for a parabola from its x and y intercepts? I have a pair of points at my disposal. One of these points represents the parabola's maximum y-value, which always occurs at x=0. I also have a point which represents the parabola's x-intercept(s). ... 2answers 363 views ### Equation for getting the length of the minor axis (of an ellipse) I'm looking for an equation that can help me determine the length of the minor axis. I know the length of the major axis and have the Cartesian coordinates of a point somewhere on the ellipse. How ... 2answers 964 views ### Find equation for hyperbola Just taking (failing) a simple algebra class, can't figure this one out and no one can explain it to me and the book just tells me to do it. Find an equation for the hyperbola described: foci ... 1answer 377 views ### Where can I find good resources to practice quadric surfaces and conics? I need to brush up my knowledge about quadric surfaces and conics. I find myself understanding things better when I work with problems. Do you know any good textbook or website with problems about ... 1answer 142 views ### Complex Square Root I am not sure where to begin on this: Determine the images of all conic sections with a focus at the origin under the principal branch of the complex square root. I probably have to use the formula ... 2answers 1k views ### Quadratic equation of an ellipse and ellipse description I found on Mathworld ( http://mathworld.wolfram.com/Ellipse.html ) that the quadratic equation: $$ax^2 + 2bxy + cy^2 + 2dx + 2fy + g = 0$$ represent an ellipse only when, after defining:$\Delta = ... 4answers 1k views ### What Does Homogenisation Of An Equation Actually Mean? For example, if we have a conic; ax2 + 2hxy + by2 + 2gx + 2fy + c = 0 What does homogenising this equation with another line (say ax + by + c = 0 ) actually mean? As in, what are the graphical ... 2answers 353 views ### finding hyperbola asymptotes Given implicit function F(x,y) = 0, how can I find its asymptotes? EDIT: Sorry, my calculations were wrong. Here is correct function:$F(x,y)=\sqrt{(x-a)^2 + (y-b)^2} - \sqrt{(x-c)^2 + (y-d)^2} - ...
Find the equation of the ellipse circumscribing a right triangle whose lengths of it's sides are $3,4,5$ and such that its area is the minimum possible one. You may chose the origin and orientation ...