The graph of the parametric equations
x=cos(t)
y=sin(t)
meets the graph of the parametric equations
x=2+4cos(s)
y=3+4sin(s)
at two points. Find the slope of the line between these two points.
So far I've set up the system of equations
cos(t)=2+4cos(s)
sin(t)=3+4sin(s)
but it seems to be getting me nowhere.
To find the slope of the line between the two points of intersection, we need to first find the coordinates of the points.
We are given the following parametric equations:
x = cos(t)
y = sin(t)
And the other set of parametric equations:
x = 2 + 4cos(s)
y = 3 + 4sin(s)
To find the intersection points, we need to solve the system of equations:
cos(t) = 2 + 4cos(s)
sin(t) = 3 + 4sin(s)
However, as you mentioned, this system seems to be giving you trouble. The reason is that it is challenging to solve a system of trigonometric equations directly.
One way to simplify the problem is to use the trigonometric identity that relates sine and cosine:
sin^2(t) + cos^2(t) = 1
Here's a step-by-step approach to solving the problem:
1. Square both sides of the first equation in the system:
cos^2(t) = (2 + 4cos(s))^2
2. Now, recall the identity sin^2(t) + cos^2(t) = 1. Rearrange it to solve for sin^2(t):
sin^2(t) = 1 - cos^2(t)
3. Substitute the expression for cos^2(t) from step 1 into the equation in step 2:
sin^2(t) = 1 - (2 + 4cos(s))^2
4. Apply the same approach to the second equation in the system:
(3 + 4sin(s))^2 = 1 - cos^2(t)
5. Now, we have two equations involving only a single variable (either t or s). You can further simplify these equations by expanding and combining like terms. The result will be a quadratic equation in either sin(t) or sin(s).
6. Solve each of the quadratic equations, and you will obtain two possible values for sin(t) and sin(s). Note that for each solution, there will be a corresponding value for cos(t) and cos(s).
7. To find the coordinates of the intersection points, use the original parametric equations:
For each value of sin(t) and cos(t), substitute them into x = cos(t) and y = sin(t) to find the x and y values of the first point.
Similarly, for each value of sin(s) and cos(s), substitute them into x = 2 + 4cos(s) and y = 3 + 4sin(s) to find the x and y values of the second point.
8. Finally, use the slope formula to find the slope of the line between the two points:
Slope = (y2 - y1) / (x2 - x1)
since sin^2(t)+cos^2(t) = 1,
(2+4cos(s))^2 + (3+4sin(s))^2 = 1
See where that gets you.