Find elements of a matrix such that its determinant is zero

[Clandry]

Right, it's the circumcircle of the triangle formed by a,b and c. Can you figure out how to do this without using the tedious expansion of the determinant?

So it is an equilateral triangle?
That means the points lie at 120degrees apart from each other.

 

[Clandry]

Had to look up what a circumcircle is and the first link is pretty much my homework problem http://mathworld.wolfram.com/Circumcircle.html

 

[Dick]

So it is an equilateral triangle?
That means the points lie at 120degrees apart from each other.

No, the triangle is a given. $a$, $b$ and $c$ are ANY points.

Had to look up what a circumcircle is and the first link is pretty much my homework problem http://mathworld.wolfram.com/Circumcircle.html

Right. But you worked it through independently.

 

[Clandry]

No, the triangle is a given. $a$, $b$ and $c$ are ANY points.
Right. But you worked it through independently.

This is going to be a dumb question.

After I find the coefficients, A, C, D, E, and F, I will have some equation that describes a circle. Does this equation by itself satisfy the requirements of the problem, where I am asked for $[x_1; x_2]$? The equation for the circle will be an implicit equation.

 

[Dick]

This is going to be a dumb question.

After I find the coefficients, A, C, D, E, and F, I will have some equation that describes a circle. Does this equation by itself satisfy the requirements of the problem, where I am asked for $[x_1; x_2]$? The equation for the circle will be an implicit equation.

The problem says 'Describe the set'. If I were grading the problem, the description of the set as the circle that passes through the points $a$, $b$ and $c$, along with the reasons you think so would be enough. I don't think the actual equation of the circle is needed.

 

[Clandry]

The problem says 'Describe the set'. If I were grading the problem, the description of the set as the circle that passes through the points $a$, $b$ and $c$, along with the reasons you think so would be enough. I don't think the actual equation of the circle is needed.

Oh I see. Originally I thought the following would also satisfy the problem:
$[x_1 x_2]^T=u[a_1 a_2]+v[b_1 b_2]+w[c_1 c_2]$ for some arbitrary constants u,v,w, such that not all of them are zero.

 

[Dick]

Oh I see. Originally I thought the following would also satisfy the problem:
$[x_1 x_2]^T=u[a_1 a_2]+v[b_1 b_2]+w[c_1 c_2]$ for some arbitrary constants u,v,w, such that not all of them are zero.

I don't think that's the best way to describe the solution.

 

[Clandry]

I don't think that's the best way to describe the solution.

Yes, I thought it was too simple to describe it like that as I did not go about finding constants u,v,w.

For the equation of the circle, it seems I still am not finding the constants, but seems a bit more informative than what I had formerly planned on doing.

 

[Clandry]

No, the triangle is a given. $a$, $b$ and $c$ are ANY points.
Right. But you worked it through independently.

I just thought of something. Couldn't the coefficients in front of the $x_1^2$ and $x_2^2$ be zero? If so, then the determiannt expansion when set to zero would yield an equation for a line that passes through those 3 distinct points.

 

[Dick]

I just thought of something. Couldn't the coefficients in front of the $x_1^2$ and $x_2^2$ be zero? If so, then the determiannt expansion when set to zero would yield an equation for a line that passes through those 3 distinct points.

Yes, you might also give some thought to 'degenerate cases'. If the three points $a$, $b$ and $c$ are collinear then the 'circumcircle' becomes a line. And yes, the coefficient of $x_1^2$ and $x_2^2$ will be zero.