Final answer:
To find the value of 2p, you need to determine the equation of the ellipse and then find the length of the latus rectum. By finding the distances between the points where the latus rectum intersects the ellipse and the foci, we can determine that 2p = 2(sqrt(41)).
Step-by-step explanation:
To find the value of 2p, we need to determine the equation of the ellipse and then find the length of the latus rectum. Given that A(0,0) and C(3,4) are the foci, we can determine the length of the major axis. The major axis is the distance between the two foci and is equal to the sum of the distances from any point on the ellipse to the two foci. It can be calculated using the formula sqrt((x2-x1)^2 + (y2-y1)^2), where (x1, y1) and (x2, y2) are the coordinates of the two foci.
The equation of the ellipse centered at the origin is (x^2/a^2) + (y^2/b^2) = 1, where a is the length of the semi-major axis and b is the length of the semi-minor axis. Since the major axis is vertical, the equation becomes (x^2/b^2) + (y^2/a^2) = 1.
From the given vertices A(0,0) and C(3,4), we can determine that a=4 and b=3. Therefore, the equation of the ellipse is (x^2/9) + (y^2/16) = 1.
The latus rectum is the chord through either focus perpendicular to the major axis. Since the length of the latus rectum is given as 12p, we can determine that the length of the semi-latus rectum is 6p units.
Using the equation of the ellipse, we can find the coordinates of the two points where the latus rectum intersects the ellipse. By substituting y=±6p into the equation, we can solve for x. This will give us two x-values for each y-value. By finding the distances between these points and the foci, we can determine that 2p = 2(sqrt(41)). Therefore, the value of 2p is approximately 9.055.