To find the length of the chord formed by the intersection of the ellipse and the line, we need to solve the system of equations formed by the ellipse equation and the line equation.

Given:
Ellipse equation: 9x^2 + 16y^2 = 144
Line equation: 3x + 4y = 12

Step 1: Solve the system of equations
We can solve the system of equations by substituting one equation into the other. Let's solve for x in terms of y from the line equation:

3x + 4y = 12
3x = 12 - 4y
x = (12 - 4y)/3

Substitute this value of x into the ellipse equation:

9((12 - 4y)/3)^2 + 16y^2 = 144
9(4 - 8y + 4y^2/9) + 16y^2 = 144
36 - 72y + 36y^2/9 + 16y^2 = 144
36 - 72y + 4y^2 + 16y^2 = 144
20y^2 - 72y + 36 = 144
20y^2 - 72y - 108 = 0

Simplify the equation:
5y^2 - 18y - 27 = 0

Step 2: Solve the quadratic equation
Factor the quadratic equation or use the quadratic formula to find the values of y:

5y^2 - 18y - 27 = 0
(5y + 3)(y - 9) = 0

Two possible solutions:
5y + 3 = 0 or y - 9 = 0

If 5y + 3 = 0, then y = -3/5
If y - 9 = 0, then y = 9

Step 3: Find the corresponding values of x
Substitute the values of y back into the line equation to find the corresponding values of x:

For y = -3/5:
3x + 4(-3/5) = 12
3x - 12/5 = 12
3x = 12 + 12/5
3x = 60/5 + 12/5
3x = 72/5
x = 24/5

For y = 9:
3x + 4(9) = 12
3x + 36 = 12
3x = 12 - 36
3x = -24
x = -8

Step 4: Find the distance between the two points
Use the distance formula to find the distance between the two points (x, y):

Distance = √((x2 - x1)^2 + (y2 - y1)^2)
Distance = √((-8 - 24/5)^2 + (9 - (-3/5))^2)
Distance = √((-40/5 - 24/5)^2 + (9 + 3/5)^2)
Distance = √((-64/5)^2 + (45/5)^2)
Distance = √((64^2/5^2) + (45^2/5^2))
Distance =