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.
Please help!!!! I need this done soon and I am getting nowhere with this problem
the two graphs are
x^2 + y^2 = 1
(x-2)^2 + (y-3)^2 = 16
x^2-4x+4 + y^2-6y+9 = 16
but, x^2+y^2 = 1, so that means
-4x+4 + 1 -6y+9 = 16
4x+6y = -2
so the slope of the line is -2/3
No need to actually find the points.
To find the slope of the line between the two points of intersection, we first need to find the points of intersection between the two parametric curves. Let's find the values of t and s that make the two sets of parametric equations equal:
x = cos(t) = 2 + 4cos(s)
y = sin(t) = 3 + 4sin(s)
To solve for t and s, we can equate the x-components and y-components separately:
cos(t) = 2 + 4cos(s) ...(1)
sin(t) = 3 + 4sin(s) ...(2)
From equation (1), we have:
cos(t) - 4cos(s) = 2 ...(3)
From equation (2), we have:
sin(t) - 4sin(s) = 3 ...(4)
Now, let's use a trigonometric identity to simplify equations (3) and (4). We will use the identity cos^2(t) + sin^2(t) = 1.
Rearranging equation (3), we get:
cos(t) - 2 = 4cos(s) ...(5)
Squaring equation (5), we have:
(cos(t) - 2)^2 = (4cos(s))^2
cos^2(t) - 4cos(t) + 4 = 16cos^2(s)
Using the trigonometric identity cos^2(t) + sin^2(t) = 1, we can rewrite the last equation as:
1 - sin^2(t) - 4cos(t) + 4 = 16cos^2(s)
Further simplification gives us:
5 - sin^2(t) - 4cos(t) = 16cos^2(s) ...(6)
Now, let's simplify equation (4):
sin(t) - 3 = 4sin(s) ...(7)
Squaring equation (7), we get:
(sin(t) - 3)^2 = (4sin(s))^2
sin^2(t) - 6sin(t) + 9 = 16sin^2(s)
Using the trigonometric identity cos^2(t) + sin^2(t) = 1, we can rewrite the last equation as:
1 - cos^2(t) - 6sin(t) + 9 = 16sin^2(s)
Further simplification gives us:
10 - cos^2(t) - 6sin(t) = 16sin^2(s) ...(8)
Now, we have two equations (6) and (8) with two variables t and s. We can solve these equations simultaneously to find the values of t and s.
Once we find the values of t and s, we can substitute them into the original parametric equations to find the points of intersection (x, y) between the two curves.
Finally, with the coordinates of the two points of intersection, we can find the slope of the line passing through them using the formula:
slope = (y2 - y1) / (x2 - x1),
where (x1, y1) and (x2, y2) are the coordinates of the two points of intersection.