2
votes
4answers
57 views

Find the center of a circle on the x-axis with only two points, no radius/angle given

Find the center $C$ on the x-axis of the circle containing $(15,-2)$ and $(7,10)$ I can't seem to find a formula to help me solve this problem without needing the radius or the angle between the the ...
1
vote
1answer
34 views

solving system of equations(nonlinear)

I am trying to solve the following system of equations: $$\frac{kq^2}{d}=mg(L-L\cos(t))+\frac{kq^2}{r}$$ $$\sin(t)=\frac{x}{L}$$ $$r^2=(L-L\cos(t))^2+(x+d)^2$$ The parameters are: $k,L,d,q,m,g$ The ...
0
votes
1answer
32 views

For the polynomial, list each real zero and its multiplicity. Determine whether the graph crosses or touches the x-axis at each x -intercept.

For the polynomial, list each real zero and its multiplicity. Determine whether the graph crosses or touches the x-axis at each x -intercept. f(x) = (1/5)x^4(x^2 - 3) the choice 1- 0, ...
0
votes
2answers
39 views

Parabolas and axis of symmetry?

I have the parabola $$(x+y)^2 = 8(x−y)$$ and know that the axis of symmetry is $$x+y=0$$ but I know when this is the case the left hand side equals 0 but apart from that I can't see how this equation ...
1
vote
2answers
62 views

Equation $1+x^8y^4+x^4y^8-x^2y^4-x^6y^6-x^4y^2=0$

How to prove that the following equation: $$1+x^8y^4+x^4y^8-x^2y^4-x^6y^6-x^4y^2=0$$ has for solution(in real numbers): $|x|=|y|=1~$ only. Any hint would be appreciated.
0
votes
2answers
31 views

Solving for points in a plane based on line lengths and geometry

I have the following points and lines in a plane: The problem is this: Given that we know the lengths of lines A, B and C, how can we calculate the coordinates of each point a, b and c? The ...
3
votes
0answers
48 views

Solving a system of 3 variables

How to solve or what is the algorithm to solve a system of equations like this: $$\eqalign{ (x +\phantom{3} z)^2 + (y +\phantom{3} w)^2 &= 52\cr (x + 3z)^2 + (y + 3w)^2 &= 296\cr (x ...
2
votes
0answers
70 views

Quadratic equation and trig

If the quadratic equation $ax^2+bx+c=0$ has equal roots where $a, b$ and $c$ denote the lengths of the sides opposite to vertices A, B and C of a triangle ABC respectively, then find the sum of ...
0
votes
1answer
140 views

Analytical Geometry:- Circles tricky question

If two distinct chords of the circle $x^2+y^2-ax-by=0$ drawn from $(a,b)$ are divided by the $x$ axis in the ratio 2:1, prove that $a^2>3b^2$.
0
votes
0answers
30 views

What can I say about these two points… [duplicate]

Two points on the graph of $y=kx^p$ are labeled $A$ and $C$. Point $A$ has coordinates $(a,b)$, where $0<a<1$ and point $C$ has coordinates $(c,d)$, where $1<c$. If we are told that that ...
4
votes
0answers
48 views

Existence of Solutions of Two Cubic Equations in a Particular Region

If I have two cubic equations in two variables, $ax^3 + bx^2 y + cxy^2+\dots=0$ and another one with different coefficients, and I know that $(x,y)=(0,0)$ or $(1,1)$ are solutions, are there any nice, ...
3
votes
1answer
208 views

Curve through four points — simple algebra??

The motivation for this is Bezier curves. But, if you don't know what these are, you can skip down to the last paragraph, where the problem is described in purely algebraic terms. Suppose I want to ...
5
votes
5answers
347 views

For $(x+\sqrt{x^2+3})(y+\sqrt{y^2+3})=3$, compute $x+y$ .

If $(x+\sqrt{x^2+3})(y+\sqrt{y^2+3})=3$, compute $x+y$.
3
votes
2answers
47 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 ...
3
votes
2answers
81 views

Showing the equations of 3 lines are dependent

I have 3 lines: $$ A_1x + B_1y + C_1 = 0 $$ $$ A_2x + B_2y + C_2 = 0 $$ $$ A_3x + B_3y + C_3 = 0 $$ That I believe are dependent. That is, all of the intersections of any pair of two of these lines ...
0
votes
1answer
153 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, ...
1
vote
0answers
51 views

Prove that $X$ parametrizes a regular surface $M$ in $\mathbb{R}^3$ and determine for which values $p$ the curve $y$ is geodesic on $M$.

Real valued functions $f,g:\mathbb{R}_+ \rightarrow \mathbb{R}$ are $f(u) = e^{-u}$ and $g(u) = \int \sqrt{(1-e^{-2t}} dt$. Given $X:\mathbb{R}_+ \rightarrow \mathbb{R}^2$ with $X(u,v) = [f(u) \cos ...
6
votes
1answer
105 views

What surfaces in $\mathbb R^3$ are such that every planar section (with more than 1 point) has nontrivial symmetry?

In $\mathbb R^3$ , the intersection of a plane and a sphere (e.g. $x^2 + y^2 + z^2 = 1$) is either empty, a single point, or a circle. All isometries of those circles are realized by isometries of ...
2
votes
0answers
132 views

symmetries of families of polynomial functions

The family of quadratic functions $F_2(a,b,c)$, consisting of all functions of the form $f(x)=ax^2+bx+c$, has the nice property (call it P) that given any $f,g\in F_2$, there is a sequence of function ...
1
vote
1answer
243 views

What is the name of these equations?

$xy=0$ $ax +by +cxy +d=0$ $ax +by +cz +dxy +eyz +gxyz=0$ I made myself the examples, sometimes I face these equations and I do not know how to resolve them, all equations whose unknowns have ...
4
votes
2answers
332 views

Calculating Intersections of Lines and Algebraic Surfaces

For context I am developing a ray-tracer for a computer science class, and want to implement some more advanced shapes than just spheres. So while this is related to schoolwork, I'm not asking you to ...