1
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1answer
19 views

need explanation of what exactly is a directrix & focus?

((I'm not asking why do we need to know conic sections etc.) Like other similar questions.) I actually love math & currently learning conic sections in class, neither my textbook or teacher ...
0
votes
2answers
36 views

how do i write an equation in standard form by completing the square for $x^2 -9y^2-4x-18y=14$

I'm really having trouble with completeing squares i can solve for circles and ellipses but i can't seem to understand hyperbolas or parabolas, help would be deeply appreciated.
0
votes
3answers
28 views

Solve a system of equations involving two ellipses

Problem #38 asks us to solve the system using either graphing, substitution, or elimination. The only way that I can think of doing this is by graphing. However, is there any easy way to solve this ...
0
votes
1answer
31 views

Show that endpoint of a focal chord is $$\left(\frac{4p^2}{x_0}, \frac{p^2}{y_0}\right)$$

If $PQ$ is a focal chord of the parabola $x^2=4py$ and the coordinates of $P$ are $(x_{0}, y_{0})$ show that the coordinates of $Q$ are $$\left(\frac{4p^2}{x_0}, \frac{p^2}{y_0}\right)$$ I labeled ...
2
votes
4answers
55 views

Find the equation of a circle passing three points (conics)

Problem: Determine the equation of the circle that passes through three points, $J(-3, 2)$, $K(4, 1)$, and $L(6, 5)$. I thought of using systems like so: $$\left\{ \begin{array}{rcl} (x+3)^2 + ...
1
vote
2answers
38 views

Conic Sections Question - Hyperbolas & Circles

So, if you have a hyperbola with foci at $(4,0)$ & $(-2,0)$, and the slopes of the asymptotes are $+4$ and $-4$, what would the equation for this hyperbola be? I know that the center would be ...
1
vote
1answer
56 views

Non-linear systems help!

I have a non-linear system of equations, $$\left\{ \begin{array}{rcl} x^2 - xy + 8 = 0 \\ x^2 - 8x + y = 0 \\ \end{array} \right.$$ I have tried equating the expressions (because both equal 0), which ...
0
votes
1answer
52 views

Show that lines created by certain points on the parabola intersect at the directrix?

Edit: I got the answer by finding points of intersection between the line passing through B and the focus and the parabola, but it didn't seem like the best solution. Any other ideas? The Segments ...
1
vote
3answers
82 views

General form of a circle

My math teacher taught me that the general form (equation) of a circle is: $$ ax^2+by^2+cx+dy+e=0 $$ He also asked us this: If the product of $c$ and $d$ is negative, then what 2 quadrants can the ...
2
votes
0answers
51 views

An Easier way to solve simple equations of this type

Im currently working with ellipses and I've been given two points on a ellipse whose major axis is along the x-axis, $(4,3)$ and $(-1,4)$. The question asks me to find the length of the major and ...
0
votes
1answer
444 views

How can convert the general form of ellipse equation in the standard form?

How can convert the general form of ellipse equation in the standard form? $$-x+2y+x^2+xy+y^2=0$$ Thank you in advance?
1
vote
0answers
68 views

How to see and proof that the hyperbola as a constant difference of distances holds for $\frac{1}{x}$?

I understand that a hyperbola can be defined as the locus of all points on a plane such that the absolute value of the difference between the distance to the foci is $2a$, which is the distance ...
-1
votes
1answer
75 views

Are Parabolas similar intuitively?

All parabolas are similar, but are they all similar in that it is just a question of 'zooming in and out' intuitively speaking? It seems that there should therefore be on all parabolas a curve from ...
2
votes
1answer
53 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 ...
0
votes
1answer
50 views

What is the vertex of this parabola and it's min value?

Th equation of the parabola is $$2\left(x+\dfrac34\right)^2−\dfrac{25}8$$ What is the vertex and the min value? and do I just plug $x$ values into the equation to get the points on the graph?
1
vote
1answer
45 views

How do I get the equation for this parabola in standard form?

How do I get the equation for this parabola in standard form? $ y = f(x)= 2x^2+3x-2$
1
vote
3answers
106 views

How to find the point on a parabola where x and y are equal?

On a parabola how could i find the point at which the y and x points are equal and meet on a point of the graph, algebraically?
2
votes
1answer
1k views

How to derive the equation of a parabola given a focus and a directrix not parallel to the x or y axis?

I was wondering if it is possible to derive a general form of a parabola given any focus and directrix. So far all the materials I have come across only show the derivation for a parabola equation ...
1
vote
2answers
549 views

coefficients of quadratic function?

In a quadratic function: coefficient $a$ controls the speed of increase/decrease from the vertex. coefficient $b$ controls the downward slope as the function crosses the y-axis. I don't really ...
1
vote
4answers
255 views

Parabola and Circle problem : The parabola $y =x^2-8x+15$ cuts the x axis at P and Q. A circle is drawn …

Problem : The parabola $y=x^2-8x+15$ cuts the x axis at P and Q. A circle is drawn through P and Q so that the origin is outside it. Find the length at a tangent to the circle from O. My approach ...
1
vote
2answers
58 views

If $A=(-4,0)$ and $B=(4,0)$, what is the locus of points $P$ such that $|AP-BP|=16$? Does it even exist?

I am stuck in this question for about a week: If there are points $A$ and $B$ such that $A(-4,0)$ and $B(4,0)$ then what is the locus of points $P$ such that $|AP-BP|=16$? I think this is a ...
3
votes
1answer
44 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
2answers
64 views

Given an semi-ellipse inscribed about a square, how do I find the equation of the ellipse?

Given the following diagram: Where: W = (-1, 0) X = (-1, 2) Y = (1, 2) Z = (1, 0) How can I find M? The ellipse can be assumed to be a semi-ellipse with one of the foci on $\bar{XY}$. I'm ...
3
votes
2answers
43 views

What is the rationale for the factor of $4$ in the Conics parabola equation?

The Conics form of a parabola equation is $4p(y-k)=(x-h)^2$ where $(h,k)$ is the vertex of the parabola and $p$ is the distance from the vertex to the focus. (Which is also the same distance from the ...
1
vote
1answer
100 views

Turning an ellipse into a parabola

Today I was discussing circles, ellipses, hyperbolas, and parabolas in my precalculus class. We did the usual: completing the square, finding the center and radius (radii), etc. etc. But I like to ...
2
votes
3answers
69 views

Find at least two ways to find $a, b$ and $c$ in the parabola equation

I've been fighting with this problem for some hours now, and i decided to ask the clever people on this website. The parabola with the equation $y=ax^2+bx+c$ goes through the points $P, Q$ and $R$. ...
0
votes
0answers
114 views

Generating parabola from points applet

Does anyone know of an applet or something that generates a parabola (graph and/or equation) given three (unique, non-colinear) points? I'm going to be mentioning this fact to my students as an aside ...
0
votes
1answer
103 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, ...
0
votes
2answers
74 views

Finding An Equation For A Parabola

The information given in this particular problem: Axis is parallel to y-axis; graph passes through and $(4,11)$.$(3, 4)$ $(0,3)$ From this information, I know that it opens either upwards or ...
1
vote
2answers
154 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
1answer
144 views

An ellipse in the rhombus

Suppose that there is an ellipse that meets with the square, but exactly inside the rhombus. The rhombus's side would be some $x$ cm. (for e.g., we can take it as $2 \ cm$.) The ellipse would have a ...
2
votes
1answer
713 views

How to calculate minimum distance between two arbitrary ellipses in 2D?

Arbitrary ellipses means that they can be scaled, translated and rotated in any way in 2D. Do you know some high-school method (might be slightly more advanced than that) to find the minimum distance? ...
1
vote
1answer
256 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 ...
1
vote
3answers
69 views

Describing the effect on $ax^2$ by manipulating $a$

Please take, for example, $y = x^2$ and $y = 2x^2$. Graphs: Wolfram Alpha What is the most appropriate way to describe the effect of $a$? "$a$ causes the parabola to open at $1/a$ the rate of $y = ...
8
votes
3answers
530 views

If two parabolas don't intersect, is there a line that doesn't intersect either of them?

Two parabolas in a plane are given, such that they don't intersect. Is it true that there is a line in plane such that doesn't intersect any of them?
2
votes
2answers
245 views

Parabola Question: How to derive an appropriate equation based on specific criteria?

I've seen similar questions, but none worded quite like what I'm asking here. I'm not a math major, so I may have follow-up questions if answers befuddle me. Scenario: I'm trying to emulate a ...
2
votes
3answers
270 views

Find the smallest $k\in \mathbb{Z}$ such that $f(x)=x^2-3x+k$ and $g(x)=x-2$ do not intersect.

$$ \large{ \text{Here are some instructions from the original question: } \ \\ } $$ $$ \large{ k \in \mathbb{Z} \ \ \land \\ f(x)=x^{2}-3x+k \ \ \land \ \ g(x)=x-2 \ \\ \text{And, there is NO any ...
1
vote
1answer
184 views

Trajectory Problem with Parabola

The height of an object is given by $$ h(x) = -0.005x^2 + x. $$ When does the object hit the ground? When does it attain its maximum height? What is its maximum height? I divided -.005/-1 to get ...
1
vote
1answer
533 views

Find intersection(s) between parametrized parabola and a line

I'm trying to find the value(s) of the parameter $t$ at the intersection point(s) between a 2D general parabola (as a parametric function of $t$) and a line whose equations can be derived from two ...
1
vote
2answers
655 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 ...
3
votes
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...
2
votes
1answer
213 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 ...
2
votes
2answers
510 views

A hyperbola as a constant difference of distances

I understand that a hyperbola can be defined as the locus of all points on a plane such that the absolute value of the difference between the distance to the foci is $2a$, the distance between the two ...
4
votes
2answers
1k 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 ...
4
votes
5answers
583 views

Usefulness of Conic Sections

Conic sections are a frequent target for dropping when attempting to make room for other topics in advanced algebra and precalculus courses. A common argument in favor of dropping them is that ...
6
votes
12answers
9k views

Derivation of the formula for the vertex of a Parabola

I'm taking a course on Basic Conic Sections, and one of the ones we are discussing is of a parabola of the form $y = a x^2 + b x + c$ My teacher gave me the formula: $x = -\frac{b}{2a}$ as the $x$ ...