Find the sum of all possible values of the constant k such that the graph of the parametric equations x=2+4cos(s) and y=k-4sin(s) intersects the graph of the parametric equations x=1+cos(t) and y=-3+sin(t) at only one point.
Find the sum of all possible values of the constant k such that the graph of the parametric equations x=2+4cos(s) and y=k-4sin(s) intersects the graph of the parametric equations x=1+cos(t) and y=-3+sin(t) at only one point.
To find the sum of all possible values of the constant k, we need to determine the values of k for which the two given parametric equations intersect at only one point.
First, let's equate the x-components and y-components of the two equations:
Equating the x-components:
2 + 4cos(s) = 1 + cos(t) ---(1)
Equating the y-components:
k - 4sin(s) = -3 + sin(t) ---(2)
To simplify equation (1), rearrange it to isolate cos(t):
cos(t) = 2 + 4cos(s) - 1 ---(3)
Similarly, rearrange equation (2) to isolate sin(t):
sin(t) = k - 4sin(s) + 3 ---(4)
Square both sides of equations (3) and (4) to eliminate the trigonometric functions:
cos^2(t) = (2 + 4cos(s) - 1)^2 ---(5)
sin^2(t) = (k - 4sin(s) + 3)^2 ---(6)
Since sin^2(t) + cos^2(t) equals 1, we can add equations (5) and (6):
(2 + 4cos(s) - 1)^2 + (k - 4sin(s) + 3)^2 = 1 ---(7)
Now we need to express equation (7) in terms of either sin(s) or cos(s) to eliminate the s parameter. Let's express it in terms of sin(s).
Substitute cos(s) = sqrt(1 - sin^2(s)) into equation (7) to get:
(2 + 4sqrt(1 - sin^2(s)) - 1)^2 + (k - 4sin(s) + 3)^2 = 1
Expand and simplify the equation:
(1 + 4sqrt(1 - sin^2(s)))^2 + (k - 4sin(s) + 3)^2 = 1
1 + 8sqrt(1 - sin^2(s)) + 16(1 - sin^2(s)) + k^2 - 8ksin(s) + 16sin^2(s) + 6k - 24sin(s) + 9 = 1
16sin^2(s) + 16 - 16sin^2(s) + 8sqrt(1 - sin^2(s)) + 9 + k^2 - 8ksin(s) + 6k - 24sin(s) = 1
8sqrt(1 - sin^2(s)) + k^2 - 8ksin(s) - 24sin(s) + 6k + 24 = 0
Setting the above equation equal to zero, we have:
8sqrt(1 - sin^2(s)) - 8ksin(s) - 24sin(s) + k^2 + 6k + 24 = 0
To find the values of k for which the parametric equations intersect at only one point, we need to ensure that this equation has exactly one real solution for sin(s) (since sin(s) varies between -1 and 1).
For a quadratic equation to have exactly one real solution, the discriminant must be equal to zero. Therefore, set the discriminant of the equation equal to zero:
(-8k)^2 - 4(8)(k^2 + 6k + 24) = 0
64k^2 - 32k^2 - 192k - 768 = 0
32k^2 - 192k - 768 = 0
Divide the equation by 32:
k^2 - 6k - 24 = 0
Factor the quadratic equation:
(k - 8)(k + 3) = 0
Setting each factor equal to zero and solving for k, we get:
k - 8 = 0 --> k = 8
k + 3 = 0 --> k = -3
Therefore, the possible values of the constant k that satisfy the given conditions are k = 8 and k = -3.
To find the sum of all possible values, add the two values:
8 + (-3) = 5
Hence, the sum of all possible values of the constant k is 5.
To find the sum of all possible values of the constant k, we need to find the values of k that satisfy the condition of having exactly one intersection point between the two parametric equations.
Let's start by equating the x-coordinates and y-coordinates of the two parametric equations:
For the x-coordinate:
2 + 4cos(s) = 1 + cos(t) ----(Equation 1)
For the y-coordinate:
k - 4sin(s) = -3 + sin(t) ----(Equation 2)
To find the values of s and t, we can solve these equations simultaneously. First, let's solve for t in terms of s using Equation 2:
sin(t) = -3 - k + 4sin(s) ----(Equation 2')
Now, let's substitute this value of sin(t) into Equation 1:
2 + 4cos(s) = 1 + cos(t) = 1 + cos(arcSin(-3 - k + 4sin(s)))
Using the identity cos(arcSin(x)) = sqrt(1 - x^2), we can simplify the equation further:
2 + 4cos(s) = 1 + sqrt(1 - (-3 - k + 4sin(s))^2)
Expanding the square root and simplifying:
2 + 4cos(s) = 1 + sqrt(1 - 9 - 6k + 16sin(s) + 9k^2 - 24ksin(s) + 16sin(s)^2)
Combining like terms:
2 + 4cos(s) = 1 + sqrt(-8 - 6k + 16sin(s) + 9k^2 - 24ksin(s) + 16sin(s)^2)
To have exactly one intersection point between the two parametric equations, we need this equation to have exactly one solution for cos(s). To achieve that, the expression inside the square root must be zero.
Therefore, we have the following equation:
-8 - 6k + 16sin(s) + 9k^2 - 24ksin(s) + 16sin(s)^2 = 0
This is a quadratic equation in sin(s) with respect to k. To find the sum of all possible values of k, we need to find the values of k that make this quadratic equation have exactly one solution. To do so, we need the discriminant (b^2 - 4ac) of the quadratic equation to be equal to zero.
The discriminant is given by:
(b^2) - 4(a)(c) = (-24k)^2 - 4(16)(-8 - 6k + 9k^2) = 576k^2 + 256(9k^2 - 6k - 8)
Setting the discriminant equal to zero:
576k^2 + 256(9k^2 - 6k - 8) = 0
Simplifying and combining like terms:
576k^2 + 2304k^2 - 1536k - 2048 = 0
2880k^2 - 1536k - 2048 = 0
Now, we can solve this quadratic equation for k. Using the quadratic formula:
k = (-b ± √(b^2 - 4ac)) / (2a)
k = (1536 ± √(1536^2 - 4(2880)(-2048))) / (2(2880))
Calculating the value inside the square root:
√(1536^2 - 4(2880)(-2048)) ≈ 2889.83
Now, substitute this back into the equation for k:
k = (1536 ± 2889.83) / (5760)
k ≈ -0.221 or k ≈ 0.568
Therefore, the sum of all possible values of the constant k is approximately -0.221 + 0.568 = 0.347.