Giving the Equation of a Line in terms of X, y and theta

[MohammedKhan]

Given an equation for a line y=7x/3, how can i define it in mathematica using x,y and theta.
I am eventually going to have to find the intersection of this line to a circle with the equation Circle[{0,9},5] as a function of theta.

I thought maybe if i use the rule tan(θ) = y/x then id get something along the lines of y = (7/3)tanθ, but then I don't know how to define this in mathematicas language.

 

[Bill Simpson]

In[1]:= Reduce[{y==7x/3,x^2+(y-9)^2==5^2},{x, y}, Backsubstitution->True]

Out[1]= (x==(189-3*Sqrt[721])/58 && y==(441-7*Sqrt[721])/58) || (x == (189+3*Sqrt[721])/58 && y == (441+7*Sqrt[721])/58)

In[2]:= N[Reduce[{y==7x/3,x^2+(y-9)^2==5^2},{x,y},Backsubstitution->True]]

Out[2]= (x==1.86975&&y==4.36276)||(x==4.64749&&y==10.8441)

 

[MohammedKhan]

in the question however it states that the beam passes through the circle as function of theta. so can i say this in addition to wat you wrote.

In[1]:= Reduce[{y==(7x/3)Tan[θ],x^2+(y-9)^2==5^2},{x, y}, Backsubstitution->True]

would this now solve for the intersection as a function of theta?

 

[Bill Simpson]

I do not understand what the question is, but I assume your guess is incorrect.

 

[Bill Simpson]

Given an equation for a line y=7x/3, how can i define it in mathematica using x,y and theta.
I am eventually going to have to find the intersection of this line to a circle with the equation Circle[{0,9},5] as a function of theta.

Is this a homework problem?
Is the problem stated exactly as you have said?
Is the problem stated in English?
Is there any perhaps simpler previous example problem with a solution available?

If you can provide all that then perhaps we can find a solution for you.

 

[MohammedKhan]

ok so here's the question

A lighthouse is located 9 miles south of the center from the center of a circular island of diameter 10 miles in d.When the beam is over the island the beam enters at a point on the shore and leaves at another point,. As the beam rotates the distance between these two points changes.

The beam is making 3 revolutions per minute clockwise.

At what rate is the the distance between the two points changing when the beam enters 3miles east of the line between the lighthose and the center of the island.

its divided into a few parts.

1. Give the equation for the beam it should be in terms x,y and tan(\[Theta]). Name it eqbeam.

2. Find the line segment between the entrance and the exit point as a function of \[Theta]. Name it lineseg

3. Give the derivative of the distance between the two points with repect to \[Theta]

4. What is the angular velocity of the beam?

5. What are the coordinates of the entry point three miles east of the line between the light house and the center of the island.

6. At what rate is the the distance between the two points changing when the beam enters 3 miles east of the line between the lighthose and the center of the island.ok the tan(theta) thing i suggested def didnt make sense i was just guessing. I am thinkin how can you make a formula that accounts for the beam rotating clockwise. i would assume the dtheta/dt = 6pi per minute. i was thinkin along the lines of (x',y') = ((xcostheta - ysintheta),(xcostheta + ysintheta)) for two arbitrary points and then using getting a line from that?

 

[MohammedKhan]

ok so i used tan[theta] = y/x to get y =xtan[theta] for the equation of the line. i think it worked out. does that make sense tho? does that form a radial line originating from the center.