Let a tangent to the curve 9x2+16y2=144 intersect the coordinate axes at the points A and B . Then, the minimum length of the line segment AB is ______

The given equation is of an ellipse, which can be rewritten as (x^2/4) + (y^2/16) = 1.

The equation of the tangent to the ellipse in the slope form is y = mx + sqrt(16m^2 + 4).

The x-intercept of the tangent line (point A) is -sqrt(16m^2 + 4)/m and the y-intercept (point B) is sqrt(16m^2 + 4).

The distance AB can be found using the distance formula, which is sqrt((x2-x1)^2 + (y2-y1)^2).

Substituting the x and y intercepts into the distance formula gives AB = sqrt((sqrt(16m^2 + 4)/m + sqrt(16m^2 + 4))^2).

Simplifying this gives AB = 2sqrt(16m^2 + 4).

To find the minimum length of AB, we need to find the minimum value of the function 2sqrt(16m^2 + 4).

This function has a minimum value when its derivative is equal to zero.

The derivative of 2sqrt(16m^2 + 4) is (32m)/(sqrt(16m^2 + 4)).

Setting this equal to zero and solving for m gives m = 0.

Substituting m = 0 back into the equation for AB gives AB = 2sqrt(4) = 4.

Therefore, the minimum length of the line segment AB is 4 units.