# Tagged Questions

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

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### Given $\Vert \vec{u} \Vert$ and $\Vert \vec{v} \Vert$ and $\angle 120^\circ$ find volume with sides $\vec{u} \times \vec{v}$, $\vec{u}$ and $\vec{v}$

I am given the following problem: Knowing that $\Vert \vec{u} \Vert = 3$ and $\Vert \vec{v} \Vert = 4$ and also $\angle (\vec{u}, \vec{v}) = 120^\circ$ find the volume of the parallelepiped with ...
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### If $x^2=\lambda$, then find the value of $\lambda$

A Circle $C_1$ is drawn having any point $P$ on $X$- axis as its centre and passing through the centre of the circle $C: x^2+y^2=1$. A common tangent to $C_1$, and $C$ intersects the circle at ...
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### Sum of slopes between points of concurrent normals to hyperbola is zero

Let $A_1, A_2, A_3, A_4$ be four points on the hyperbola $xy = 1$. Suppose that the normals to the hyperbola at these four points are concurrent, i.e. they intersect in a single point. Prove that ...
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### Show that, if $u,v,w$ are orthogonal two-by-two, then $S = \{ u , v , w\}$ forms a basis which is linearly independent

I am given the following question: Show that, if $u,v,w$ are orthogonal two-by-two, then $S = \{ u , v , w\}$ forms a basis which is linearly independent. My idea to tackle this problem is to ...
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### intersection of a line (certain direction) and a circle

I need to calculate (previously) the point where a ball will touch the inside of a circle (for a game I'm developing). So I have two equations, one of the direction of the ball, and another of the ...
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### Shortest distance between parallel line and plane

I've been doing questions regarding the shortest distance between lines/planes and points , and I've come across a question asking to find the shortest distance between a line and a plane which are ...
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### Dimensions of bounding box for arbitrary circle sector

I need to determine the dimensions of bounding box for arbitrary circle sector as shown in the diagram below. Given: φ = Start angle in the range of 0 ~ 2π θ = Sweep angle in the range of 0 ~ 2π r =...
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### Largest bifocal triangle in an ellipse

Ellipse $E$ has foci at $P$ and $Q$, and semi-major and semi-minor axes of length $a$ and $b$, respectively. Find the area of the largest triangle that can be (parttially) inscribed in ellipse $E$, ...
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### Volume calculation with change of variables

I am trying to calculate the volume of a solid, given the equations of its bounding surfaces. It is a $3$-dimensional object, so the equations are in $x$, $y$ and $z$. In order to simplify the ...
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### Analytical geometry - Finding the coordinates of point M

I've been practicing analytical geometry lately and I've come to a problem. I solved the problem a few times but I can't get the right result. Here is the math problem: Point M whose distance ...
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### Congruent number $23$

The set of Congruent numbers are all the integer areas of rational sided right triangles. This means that if g is a Congruent number there exists some integer $n$ such that $g \cdot n^2$ is the ...
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### Show that pair of straight lines $ax^{2}+2hxy+ay^{2}+2gx+2fy+c=0$… meet coordinate axes in concyclic points.Also find equation of

Show that pair of straight lines $ax^{2}+2hxy+ay^{2}+2gx+2fy+c=0$ meet coordinate axes in concyclic points. Also find equation of the circle through those cyclic points My Attempt Given equation ...
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### Calculate equally-spaced and arranged center points on the edges of a circle based on its diameter and grid origin?

Question for a project: I have a circle that I know the diameter of (and therefore also the height/width dimensions of on a grid based on its top/left origin X/Y)... but I wanted to calculate equally ...
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### Ellipse and parallel lines

Let's imagine that we have an ellipse described by the known equation $v^TAv=0$, (Link_1) where $v=[x \ y \ 1]^T$ (it can be a skew one in a general case). Then we have all possible parallel lines - ...
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### graph of $z=2x+y$

The usual technique using traces where one variable is set to 0 does not seem to work here since I get all 0's and so where do the intersections meet? I looked in my calc. book and the technique ...
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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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### Equations of the tangent planes to the sphere

Find the equations of the tangent planes to the sphere $x^2+y^2+z^2-10x+2y+26z-113=0$ which are parallel to the straight lines $\frac{x+5}{2}=\frac{y-1}{-3}=\frac{z+13}{2}$ and \$\frac{x+7}{3}=\frac{y+...