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Questions tagged [analytic-geometry]

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6 votes
6 answers
1k views

why or when do we use the general form of conic sections?

why write a conic in this form? for any real A, B, C, D, E, and F, a conic may be expressed as $$Ax^2 + Bxy +Cy^2+Dx+Ey+F=0$$ why wouldn't it be more natural to write the conic in standard form with ...
Lenny's user avatar
  • 1,068
2 votes
2 answers
178 views

Comparison of two ways to introduce translation to 12-14 year olds

I consider pupils 12-14 years old, who are new to translation. On the other hand, they have been accustomed to placing points in a coordinate system, especially when studying relative numbers. In ...
Stéphane Jaouen's user avatar
1 vote
3 answers
771 views

Equation of a straight line on two dimensional Cartesian plane

What is the most appropriate way to write equation of a straight line through a given point $(\alpha,\beta)$? If you write it using the standard form $y- \beta = m (x - \alpha )$ Where m is the ...
Janaka Rodrigo's user avatar
7 votes
7 answers
685 views

Creative problems in 2D vector geometry

What are some "interesting" and creative problems or exercises on specifically 2-dimensional vector geometry that a high school student might find compelling to solve? The class' current ...
Damian Reding's user avatar
2 votes
5 answers
947 views

When are students taught implicit and parametric representations of curves?

Do students learn implicit equations (such as $x^2+y^2-r^2 = 0$) and parametric equations (e.g., $x=a t^2,\;y= 2 a t$) in a first course in algebra, which in the US would be early high school, maybe ...
Joseph O'Rourke's user avatar
2 votes
1 answer
63 views

On a special degenerate conic

I have a question on MSE that maybe can be better posed here. The question is about degenerate conics, and especially the case of two parallel lines, as in the equation $ 𝑥^2+2𝑥𝑦+𝑦^2=1$. ...
Emilio Novati's user avatar
7 votes
5 answers
887 views

How do you plausibly explain that the geometric and the coordinate expressions for the scalar product are equivalent?

The standard scalar product on $\mathbb{R}^3$ is defined via $$\vec a\cdot\vec b := a_1b_1+a_2b_2+a_3b_3$$ On the other hand, it can be expressed in a more geometrical way through the lengths of the ...
Photon's user avatar
  • 602
6 votes
3 answers
258 views

Tips for choosing coordinates of three points such that the coordinates of the orthocenter are integers

I want to give my students the coordinates of three points and ask them to find the coordinates of the orthocenter. This is a fairly long problem that can involve finding the equations of the ...
Chaim's user avatar
  • 675
15 votes
7 answers
3k views

How do you explain why perpendicular lines have negative reciprocated slopes?

For my purposes, I am interested mostly in a medium-sized liberal art college setting. My students have mostly seen this before, but it is not something they understand. When discussing parallel lines,...
David Steinberg's user avatar
4 votes
0 answers
119 views

Making co-ordinate geometry interesting for XI grade students

I am presently teaching eleventh grade (XI standard) students an introductory course in co-ordinate geometry with a focus on preparations for competitive exams. I have seen books like S.L.Loney's co-...
Spai's user avatar
  • 309
12 votes
3 answers
545 views

Hands-on demonstration ideas for multivariate calculus

In teaching Calculus III geometry plays a very important role. It is crucial that students get a good sense of how to visualize curves, surfaces, coordinate axis, frames to curves, vector fields and ...
James S. Cook's user avatar