The equation y = a^x represents an exponential function where a is a constant.

To determine the nature of the lines, we can analyze the discriminant of the quadratic equation.

The lines are perpendicular if the product of their slopes is -1. We can find the slopes from the equation and check this condition.

The discriminant is negative, indicating that the lines are perpendicular.

The lines intersect at the origin (0,0) as derived from the equation.

The lines intersect at the origin, which can be verified by substituting x = 0 and y = 0 into the equation.

The lines are perpendicular if the product of their slopes is -1, which can be verified from the equation.

The lines intersect at the origin and are not parallel, hence they are intersecting.

Completing the square shows that the lines intersect at two distinct points.

The equation can be factored as (x - 3y)^2 = 0, indicating that the lines are coincident.

The slopes can be found by solving the quadratic equation in terms of m, yielding slopes -1 and -2.

The discriminant of the quadratic equation is positive, indicating two distinct real roots.

The product of the slopes of the lines is given by m1*m2 = c/a = 1/2 = -2.

Factoring gives (2x - 3y)(2x - 3y) = 0, indicating the lines are coincident.

The discriminant of the quadratic equation is positive, indicating two distinct real roots.

The discriminant of the quadratic equation is negative, indicating imaginary lines.

Rearranging gives (x-2)^2 + (y-2)^2 = 0, which represents two intersecting lines.

To determine the nature of the lines, we can rewrite the equation in the form of (x - a)^2 + (y - b)^2 = r^2 and analyze the discriminant.

Rearranging gives (x-2)^2 + (y-3)^2 = 0, which represents a single point, hence the lines are coincident.

To determine the nature of the lines, we can find the slopes from the equation. The product of the slopes will help us conclude if they are perpendicular.

The given equation can be factored as (x - 2y)(x - 2y) = 0, indicating that the lines are perpendicular.

Since the coefficient of (x - 2)^2 is negative, the parabola opens downwards.

Rearranging gives y = (2/3)x + 2, so slope = 2/3.

Rearranging gives y = (3/4)x + 3. Slope = 3/4.

The slopes can be found by solving the quadratic equation derived from the given equation.

Using the quadratic formula, the slopes are found to be -3/5 and -2/5.

The slopes can be calculated using the quadratic formula, yielding -3/5 and -2/5.

The vertices of the ellipse 9x^2 + 16y^2 = 144 are located at (±3, 0).

The slopes of the lines are -2/3 and 4. The angle θ can be found using tan(θ) = |(m1 - m2) / (1 + m1*m2)|.

Using the formula for the angle between two lines, we find that the angle is 90 degrees.