To solve the problem, we will use the distance formula to represent the relationship between the epicenter and the station.
a. Equation of the curve:
Let's assume the epicenter's coordinates on the curve are (x, y). Since we know that the seismological station is located at (0, -4) and the epicenter is 10 kilometers away from the station, we can use the distance formula:
β((x - 0)^2 + (y - (-4))^2) = 10
Simplifying this equation gives us:
β(x^2 + (y + 4)^2) = 10
To find the equation of the curve, we square both sides of the equation:
x^2 + (y + 4)^2 = 10^2
Thus, the equation of the curve that contains the possible location of the epicenter is:
x^2 + (y + 4)^2 = 100
b. Coordinates of the epicenter:
If the epicenter is 7 kilometers away from the shore, we can modify the equation above.
Since the x-axis is the shoreline, the y-coordinate of any point on the shoreline is 0. Since the epicenter is 7 kilometers away from the shore, we can replace y with 0 in the equation.
x^2 + (0 + 4)^2 = 100
x^2 + 16 = 100
x^2 = 100 - 16
x^2 = 84
Taking the square root of both sides:
x = Β±β84
Therefore, the coordinates of the epicenter are (β84, 0) and (-β84, 0).