Step 1: Understand the geometric condition. If two parallel lines are both tangent to a circle, the centre is equidistant from each, so it lies on the line exactly midway between them. Step 2: Write both tangents with the same coefficients. The first is $3x-4y+4=0$. Dividing $6x-8y+7=0$ by $2$ gives $3x-4y+\dfrac72=0$. They are parallel. Step 3: Take the midway line. The locus is $3x-4y+k=0$ where $k$ is the average of the two constants. Step 4: Average the constants. $k=\dfrac{4+\frac72}{2}=\dfrac{\frac{15}{2}}{2}=\dfrac{15}{4}$. Step 5: Write the locus. So the locus is $3x-4y+\dfrac{15}{4}=0$. Step 6: Clear the fraction. Multiplying by $4$: $12x-16y+15=0$. \[ \boxed{12x-16y+15=0} \]