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.
To find the slope of the line between the two points where the given parametric equations intersect, we need to find the values of t and s for which the equations are equal.
We can see that x = cos(t) and x = 2 + 4cos(s). Since these are equal, we can set up the following equation:
cos(t) = 2 + 4cos(s) ...(1)
Similarly, y = sin(t) and y = 3 + 4sin(s). Setting them equal gives:
sin(t) = 3 + 4sin(s) ...(2)
Now, we can solve equations (1) and (2) simultaneously to find the values of t and s when the two parametric equations intersect.
From equation (1), we can isolate cos(t) by subtracting 2 from both sides:
cos(t) - 2 = 4cos(s) ...(3)
Similarly, from equation (2), we can isolate sin(t) by subtracting 3 from both sides:
sin(t) - 3 = 4sin(s) ...(4)
Now, we square both sides of equations (3) and (4) to eliminate the trigonometric functions:
(cos(t) - 2)^2 = (4cos(s))^2
(sin(t) - 3)^2 = (4sin(s))^2
Expanding and simplifying these equations gives:
cos^2(t) - 4cos(t) + 4 = 16cos^2(s)
sin^2(t) - 6sin(t) + 9 = 16sin^2(s)
Using the Pythagorean identity cos^2(t) + sin^2(t) = 1, we can rewrite the equation for cos(s) in terms of sin(t):
1 - sin^2(t) - 4cos(t) + 4 = 16cos^2(s)
Simplifying further:
- sin^2(t) - 4cos(t) + 3 = 16cos^2(s)
Replacing cos^2(s) with 1 - sin^2(s) using the Pythagorean identity gives:
- sin^2(t) - 4cos(t) + 3 = 16(1 - sin^2(s))
We now have a quadratic equation in terms of sin(t) and sin(s), which we can solve to find the values of sin(t) and sin(s).
Once we have the values of sin(t) and sin(s), we can substitute them back into equations (1) and (2) to find the corresponding values of cos(t) and cos(s).
Finally, we can use the coordinates (x, y) corresponding to these values to find the two points of intersection.
Once we have the coordinates of these two points, we can use the slope formula to calculate the slope of the line between them:
Slope = (y2 - y1) / (x2 - x1)
By plugging in the coordinates, we can find the slope of the line between the two points where the parametric equations intersect.