Key idea: Diameters pass through the center

Since the given lines $2x - 3y = 5$ and $3x - 4y = 7$ are diameters of the circle, both lines must pass through the center of the circle.

So, the center is the point of intersection of these two lines.

Find the center by solving the two equations

We solve: $2x - 3y = 5 \quad ...(1)$ $3x - 4y = 7 \quad ...(2)$

Multiply (1) by $3$: $6x - 9y = 15$

Multiply (2) by $2$: $6x - 8y = 14$

Subtract: $(6x-8y) - (6x-9y) = 14-15$ $y = -1$

Substitute into (1): $2x - 3(-1) = 5$ $2x + 3 = 5$ $2x = 2$ $x = 1$

Hence, the center is $C(1,-1).$

Use the area to find the radius

Area of the circle is given as $154$ square units.

$\pi r^2 = 154$

Using $\pi = \frac{22}{7},$ $r^2 = \frac{154}{\pi} = \frac{154}{22/7} = 154 \cdot \frac{7}{22} = 49$

So, $r = 7.$

Write the equation of the circle

Standard form: $(x-h)^2 + (y-k)^2 = r^2$

Here, $(h,k) = (1,-1)$ and $r^2 = 49$.

Therefore, $(x-1)^2 + (y+1)^2 = 49$

Expand the equation

$x^2 - 2x + 1 + y^2 + 2y + 1 = 49$ $x^2 + y^2 - 2x + 2y + 2 = 49$ $x^2 + y^2 - 2x + 2y = 47$

Match with the options

This matches: $\boxed{x^2 + y^2 - 2x + 2y = 47}$

So, the correct option is D.

Comparison with stored answer

Stored correct answer: D

Our derived answer is also D, so they agree.