If  y  =  mx + c  is common tangent to  x225 + y24  =  1, x2 + y2  =  16  and  y2  =  4ax(a  >  0) , then which of the following can be TRUE?focus of the parabola is at  (8√73, 0)  √3(m + c)  =  2 + 4√7  √3(m + c)  =  −2 − 4√7 Latus rectum of the parabola is  2√73

The given equations represent an ellipse, a circle, and a parabola respectively.

1. For the line y = mx + c to be a tangent to the ellipse x^2/25 + y^2/4 = 1, the condition is c^2 = 25m^2 + 4.

2. For the line to be a tangent to the circle x^2 + y^2 = 16, the condition is c^2 = 16(1 + m^2).

Equating these two conditions, we get 25m^2 + 4 = 16 + 16m^2, which simplifies to 9m^2 = 12, or m^2 = 4/3. So m = ±2/√3.

3. For the line to be a tangent to the parabola y^2 = 4ax, the condition is mc = a. Since the line is a tangent to the parabola, the value of 'a' should be positive.

Given that the focus of the parabola is at (8√73, 0), the value of 'a' is half the x-coordinate of the focus, or 4√73.

Substituting m = 2/√3 and a = 4√73 into the equation mc = a, we get c = 2√73.

So the equation of the line is y = (2/√3)x + 2√73.

Therefore, √3(m + c) = √3[(2/√3) + 2√73] = 2 + 2√219, which is not equal to 2 + 4√7 or -2 - 4√7.

The latus rectum of the parabola is 4a = 4(4√73) = 16√73, which is not equal to 2√73.

So, none of the given options can be true.