The tangent at an extremity (in the first quadrant) of latus rectum of the hyperbola x²/4 - y²/5 = 1, meets x-axis and y-axis at A and B respectively. Then (OA)² - (OB)², where O is the origin, equals:

Equation of hyperbola will be x²/4 - y²/5 = 1
So, a² = 4 and b² = 5
We know that e²a² = a² + b²
4e² = 4 + 5
e² = 9/4
e = 3/2
Latus rectum L will be (ae, b²/a)
L = (2 × 3/2, 5/2) = (3, 5/2)
We know that equation of tangent (x₁, y₁) will be xx₁/a² - yy₁/b² = 1
3x/4 - (5/2)y/5 = 1
3x/4 - y/2 = 1
X-intercept will be 4/3
Y-intercept will be -2
OA² - OB² = (4/3)² - (-2)² = 16/9 - 4 = (16 - 36)/9 = -20/9
However, there's a mistake in the above calculation. Let's correct it.
The equation of the tangent at (3, 5/2) is:
(x)(3)/4 - (y)(5/2)/5 = 1
3x/4 - y/2 = 1
To find the x-intercept (A), set y = 0:
3x/4 = 1
x = 4/3
So OA = 4/3
To find the y-intercept (B), set x = 0:
-y/2 = 1
y = -2
So OB = 2
Therefore, (OA)² - (OB)² = (4/3)² - (2)² = 16/9 - 4 = -20/9. This is not among the options.
Let's re-examine the equation of the tangent at (3, 5/2):
3x/4 - y/2 = 1
3x - 2y = 4
x-intercept: Set y=0, x = 4/3. OA = 4/3
y-intercept: Set x=0, -2y = 4, y=-2. OB = 2
(OA)² - (OB)² = (4/3)² - 2² = 16/9 - 4 = -20/9
There must be an error in the problem statement or the given options.