# Tagged Questions

Questions on the use of algebraic techniques to prove geometric theorems.

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### Graph of the function $\cos(x)\cos(x+2)-\cos^2(x+1)$ will be?

Graph of the function $\cos(x)\cos(x+2)-\cos^2(x+1)$ will be? (A)A straight line (B)A parabola Give the corresponding equation too. Source:JEE 1997. Can ...
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### Necessary & Sufficient condition for the line $ax+by+c=0$ to pass through the 1st quadrant

What is the necessary and sufficient condition for the line $ax+by+c=0$, where $a,b,c$ are non-zero real numbers, to pass through the first quadrant? I could find the points at which the line ...
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### Given points P (2,3),Q (4,-2),R (a,0) what should be the value of a if |PR-RQ| Is maximum?

Given points P (2,3),Q (4,-2),R (a,0) what should be the value of a if |PR-RQ| Is maximum ? I tried that maybe the points are collinear but I'm getting wrong answer applying collinearity condition i....
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### Small circles on sphere: finding angles for constant “cosine” onto a parallel.

My problem can be best explained starting from a 2D example: Imagine having a circle and wanting to discretize N points on the circumference of the circle so that the difference of the cosine of each ...
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### Finding an equation of a circle

My math homework are finding an equation of a circle. Given that the center is at (-10,0) and passes through A(-6,3). Second item is the given center is at (-4, 6) and is tangent to the axis. I've ...
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### General equation of a cone

What is the general equation of a cone in $\mathbb{R}^3$ space? There should be no assumptions about the location of the vertex, direction of the axis or aperture angle, these should all be variable.
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### Find the plane which touches the cone $x^2+2y^2-3z^2+2yz-5zx+3xy=0$ along the generator whose direction ratios are $1,1,1.$

Find the plane which touches the cone $x^2+2y^2-3z^2+2yz-5zx+3xy=0$ along the generator whose direction ratios are $1,1,1.$ Let the plane touches the cone at $(\alpha,\beta,\gamma)$. We know that ...
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### Find function for graph

I would like to find a function for the following graph: I have drawn the graph myself, so not all subtle bends are to be replicated. I have noted the important points the graph should have in the ...
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### Given: 2 lines containing the diameter of a circle and a point lying on this circle; Find: the equation of this circle

The lines $y = \frac{4}{3}x - \frac{5}{3}$ and $y = \frac{-4}{3}x - \frac{13}{3}$ each contain diameters of a circle. and the point $(-5, 0)$ is also on that circle. Find the equation of this ...
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### Is the flow of an analytic vector field also analytic?

Let $X$ be an analytic vector field on a smooth manifold. Is it true that the flow $\Phi_t:M\to M$ associated to that vector field is also analytic?
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### Position of Object Suspended on a String (Need Another Answer)

I'm going to try to make as few errors in typing this as possible, so please bear with me and ask me to clarify/correct whatever needed. Q: If an object is suspended on a string hung between two ...
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### How to find the equation for the circle when…

A circle goes trough two points, $A=(-1,2)$ and $B=(3,0$). You also know that the center of the circle is an element of the following linear equation: $$k \leftrightarrow 2x+y+3=0 .$$ How can you ...
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### Longest distance to the foci or the center that a point within the ellipse can be?

Given an ellipse $E$ (with the foci $f_1$ and $f_2$ and the center $c$), and a point $p$, which is the maximum distance that $p$ can be to all these 3 points to be within the ellipse $E$? I.e., which ...
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### $2$ points on a curve have a common tangent

Let $2$ points $(x_1,y_1)$ and $(x_2,y_2)$ on the curve $y=x^4-2x^2-x$ have a common tangent line. Find the value of $|x_1|+|x_2|+|y_1|+|y_2|$. It seems to me that I a missing a link and hence the ...
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### Show that the $ZX-$ plane cuts it in the curve $F(\frac{bx}{x-a},\frac{cx-az}{x-a})=0,y=0.$

The vertex of the cone is $(a,b,c)$ and $YZ$-plane cuts it in the curve $F(y,z)=0,x=0$.Show that the $ZX-$ plane cuts it in the curve $F(\frac{bx}{x-a},\frac{cx-az}{x-a})=0,y=0.$ Let the equation ...
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### Mathematical description on the interface of two adjacent bodies.

I am recently studying about a problem related to shortest path. I can briefly describe my idea but I am not sure if there is some "professional" mathematical description about it. In the following ...
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### finding a function given a slope and a point

I need to find the function $f(x)$ that is tangent to a line whose slope is given by $\displaystyle \frac{(1+\sqrt x)^{\frac{1}{2}}}{8\sqrt x}$ that passes through the point $(9,8/9)$. I really don'...
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### Find the ratio of slope

Note : Elevation $46000$ and all dimention in $mm$ (milimeter) The pipe will be installed on a surface of module structure, that module structure has different surface. I want to know " ratio ...
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### Find the range of a

If the point $P(a^2,a)$ lies in the region corresponding to the acute angles between the lines $2y=x$ and $4y=x$ then range of a is... This would have been easy if the lines had constants and I ...
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### A straight line moves so as to meet the straight lines

A straight line moves so as to meet the straight lines $y=mx, z=c$ and $y=-mx, z=-c$ in A and B and intersects the curve $yz=k^2, x=0$, show that the locus of the middle point of $AB$ is (m^...