Final answer:
To find the equations of the vertical tangent lines of the given ellipse, differentiate the equation of the ellipse with respect to x, solve for dy/dx, find the x-coordinates of the points where dy/dx is undefined, substitute the values into the ellipse equation, and write the equations of the tangent lines.
Step-by-step explanation:
To find the equations of the vertical tangent lines of the given ellipse, we need to find the points on the ellipse where the derivative of y with respect to x is undefined. These points correspond to the vertical tangent lines.
1. Differentiate the equation of the ellipse with respect to x: 18x + 2y(dy/dx) - 72 + 6(dy/dx) = 0.
2. Solve the resulting equation for dy/dx to find the derivative of y with respect to x.
3. Set the derivative equal to undefined (dy/dx = undefined) and solve for x to find the x-coordinates of the points where the vertical tangent lines intersect the ellipse.
4. Substitute the found x values into the equation of the ellipse to find the corresponding y values.
5. Finally, write the equations of the vertical tangent lines using the points found in the form y = mx + c, where m is the slope and c is the y-intercept.