Need explanation to the answers

[Atatiliri34]

Hey,

I already have the fact sheet for this question, but it only shows the answer to each sub question. Could someone show me the explanations?

All the best

 

[stapel]

It is difficult to read the tiny attachment. I think the exercise is as follows:

3. Let A and B be the following sets:
$\displaystyle A\ =\ \{(x,\ y)\ \in\ \mathbb{R}^2\ \ |\ \ y^2-x+2y+1=0\ \}$
$\displaystyle B\ =\ \{(x,\ y)\ \in\ \mathbb{R}^2\ \ |\ \ \frac{x^2}{4}+y^2=1\ \}$

a. Represent A and B on the Cartesian plane.

Consider the following set of lines:
$\displaystyle L_k\ =\ \{(x,\ y)\ \in\ \mathfrak{R}^2\ \ |\ \ y=kx\ \}$
...with parameter $\displaystyle k\ \in\ \mathbb{R}$.

b. Find the point(s) of the set B where its tangent line is orthogonal to $\displaystyle L_k$ in the case $\displaystyle k=0$.

c. Find the point(s) of the set A where its tangent line is parallel to $\displaystyle L_k$ in the case $\displaystyle k=\frac{1}{2}$.

Let's now consider the subset $\displaystyle A\cap B$.

d. Find the coordinates of the singular point(s) of the curve representing $\displaystyle A\cap B$ (that is, the points where the curve is not differentiable) which have a negative ordinate.

e. Compute the angle(s) formed between the tangent lines at this/these point(s).

Please reply with corrections, if necessary. Either way, please reply with a clear listing of your thoughts and efforts so far.

Thank you!

Eliz.

 

[blamocur]

It is difficult to read the tiny attachment. I think the exercise is as follows:



Please reply with corrections, if necessary. Either way, please reply with a clear listing of your thoughts and efforts so far.

Thank you!

Eliz.

I believe it is "-x" in A, not "-2x".

 

[blamocur]

Hey,

I already have the fact sheet for this question, but it only shows the answer to each sub question. Could someone show me the explanations?

All the best

I am confused by "curve representing $A\cap B$" : I believe $A\cap B$ is a finite set of points, not a curve. Am I missing something here?

 

[topsquark]

I am confused by "curve representing $A\cap B$" : I believe $A\cap B$ is a finite set of points, not a curve. Am I missing something here?

Set A is a parabola and set B is a circle. They intersect so there is a "curve." (But I'm not sure I'd call it that.) Anyway, there is a continuous set of points involved.

-Dan

 

[Steven G]

Set A is a parabola and set B is a circle. They intersect so there is a "curve." (But I'm not sure I'd call it that.) Anyway, there is a continuous set of points involved.

-Dan

Can you please tell me your definition of a continuous set?

 

[topsquark]

Can you please tell me your definition of a continuous set?

I just looked it up. Apparently, I'm co-opting a term that means something other than what I had thought. Sorry about that!

What I meant was that we may break the union of the sets into "small" segments and define a continuous mapping that takes us along the curve. (Something like local path continuity.) Technically speaking we actually have two of these, but since the sets intersect, I had presumed that we could still write a continuous function to get from any one point in the set to another by a continuous path.

Perhaps my Topology is off?

-Dan

 

[blamocur]

What I meant was that we may break the union of the sets into "small" segments

I thought that "$\cap$" stands for the intersection not the union.

 

[topsquark]

I thought that "$\cap$" stands for the intersection not the union.

How is it that I can talk about Topology and yet miss that??

I'm sick. Leave me alone!

-Dan