If the straight line x-y=k and the circle x^2+y^2+2x-4y-1=0 intersect at A and B, then the x-coordinate of the mid-point of AB is _ ?
A. 1+k
B. 1-k
C. (1+k)/2
D. (1-k)/2
To find the x-coordinate of the midpoint of AB, we need to find the coordinates of points A and B first.
The equation of the line is x - y = k, which can be rearranged as y = x - k.
Substituting this equation into the equation of the circle, we get:
x^2 + (x - k)^2 + 2x - 4(x - k) - 1 = 0
Simplifying this equation, we have:
2x^2 - 2(k + 1)x + k^2 + 5k - 1 = 0
Now we can solve this quadratic equation to find the x-coordinates of A and B. Using the quadratic formula, we have:
x = (-b ± √(b^2 - 4ac)) / (2a)
For our equation, a = 2, b = -2(k + 1), and c = k^2 + 5k - 1. Plugging these values into the formula, we get:
x = [-(-2(k + 1)) ± √((-2(k + 1))^2 - 4(2)(k^2 + 5k - 1))] / (2(2))
x = [2(k + 1) ± √(4(k + 1)^2 - 8(k^2 + 5k - 1))] / 4
x = [(k + 1) ± √(4k^2 + 8k + 4 - 8k^2 - 40k + 8))] / 4
x = [(k + 1) ± √(-4k^2 - 32k + 12)] / 4
Since the circle and the line intersect at two points, A and B, we will have two possible values for x: x1 and x2.
Now, let's calculate the x-coordinate of the midpoint of AB:
x-coordinate of midpoint = (x1 + x2) / 2
x-coordinate of midpoint = {[(k + 1) + √(-4k^2 - 32k + 12)] / 4 + [(k + 1) - √(-4k^2 - 32k + 12)] / 4} / 2
x-coordinate of midpoint = [(2k + 2) / 4] / 2
x-coordinate of midpoint = (1 + k) / 4
Therefore, the x-coordinate of the midpoint of AB is (1 + k) / 4.
The correct answer is not provided in the options.
To find the x-coordinate of the midpoint of AB, we first need to find the coordinates of points A and B where the line and the circle intersect.
Let's solve the given equations simultaneously to find the intersection points.
1. Line equation: x - y = k ---------- (1)
2. Circle equation: x^2 + y^2 + 2x - 4y - 1 = 0
To solve these, we can either use substitution or elimination. Let's use the substitution method:
From equation (1), we can rewrite it as: y = x - k.
Substitute this value for y in equation (2):
x^2 + (x - k)^2 + 2x - 4(x - k) - 1 = 0
x^2 + x^2 - 2kx + k^2 + 2x - 4x + 4k - 1 = 0
2x^2 - 4kx + 4k^2 - 3x + 4k - 1 = 0
2x^2 - (4k + 3)x + (4k^2 + 4k - 1) = 0
Now we have a quadratic equation. To find the x-coordinate values of A and B, we need to solve this quadratic equation. The coordinates of the intersection points will be in the form (x, y).
We can solve the quadratic equation using the quadratic formula: x = (-b ± √(b^2 - 4ac))/(2a)
In this case, a = 2, b = -(4k + 3), and c = 4k^2 + 4k - 1.
Therefore,
x = { -[-(4k + 3)] ± √((4k + 3)^2 - 4*2*(4k^2 + 4k - 1)) } / (2*2)
x = { (4k + 3) ± √(16k^2 + 24k + 9 - 32k^2 - 32k + 8) } / 4
x = { (4k + 3) ± √(-16k^2 - 8k + 17) } / 4.
Since the discriminant (-16k^2 - 8k + 17) is not guaranteed to be a perfect square, we cannot simplify the expression for x further.
Therefore, the x-coordinate of the midpoint of AB will be the average of the x-coordinates of points A and B.
x = (x_A + x_B) / 2.
Since we do not know the exact values of x_A and x_B, we cannot determine the exact x-coordinate of the midpoint.
Hence, the answer cannot be determined from the given information.
since y=x-k,
x^2 + (x-k)^2 + 2x - 4(x-k) - 1 = 0
2x^2 - 2x(k+1) + k^2+4k-1 = 0
x = [2(k+1)±√(4(k+1)^2-8(k^2+4k-1))]/4
now, that looks like a lot of complicated stuff, but notice that it's really
(k+1)/2 ± some stuff
So, (k+1)/2 is the midpoint coordinate