Equation of the Straight Line

To find the equation of the straight line joining the foci of the ellipse x^2/25 + 5y^2/16 = 1, we need to determine the coordinates of the foci first.

The standard form of an ellipse is given by (x^2/a^2) + (y^2/b^2) = 1, where a and b are the semi-major and semi-minor axes, respectively.

For the first ellipse, we have a = 5 and b = 4. The equation can be rewritten as (x^2/5^2) + (y^2/4^2) = 1. Comparing this with the standard form, we find that the semi-major axis is a = 5 and the semi-minor axis is b = 4.

The equation for the foci of an ellipse is given by c^2 = a^2 - b^2, where c is the distance from the center of the ellipse to each focus.

For the first ellipse, c^2 = 5^2 - 4^2 = 9, so c = 3.

The coordinates of the foci are given by (±c, 0), which in this case are (±3, 0).

Using these coordinates, we can find the slope of the line joining the foci using the formula m = (y2 - y1)/(x2 - x1), where (x1, y1) and (x2, y2) are the coordinates of the two foci.

Let's take the foci as A(-3, 0) and B(3, 0).

m = (0 - 0)/(3 - (-3)) = 0/6 = 0.

Since the slope is 0, the equation of the line joining the foci is y = 0.

Area of the Figure Formed by the Foci

To find the area of the figure formed by the foci of the two ellipses, we need to determine the coordinates of the foci for the second ellipse.

The second ellipse has a = 4√6 and b = 7. The equation can be rewritten as (x^2/(4√6)^2) + (y^2/7^2) = 1.

Using the formula c^2 = a^2 - b^2, we find that c^2 = (4√6)^2 - 7^2 = 96 - 49 = 47.

Taking the square root of both sides, c = √47.

The coordinates of the foci are given by (±c, 0), which in this case are (±√47, 0).

Using these coordinates, we can find the slope of the line joining the foci as we did before.

Let's take the foci as C(-√47, 0) and D(√47, 0).

m = (0 - 0)/(√47 - (-√47)) = 0/2√47 = 0.

Again, the slope is 0, so the equation of the line joining the foci is y = 0.

The figure formed by the foci of the two ellipses is a