1. Given complex numbers

   z_1=2+3i,qquad z_2=3+4i

   Let $z=x+iy$ where $x,y\in\mathbb R$.

2. Use the given condition

   The set is $S=\left\{z\in\mathbb C:\ |z-z_1|^2-|z-z_2|^2=|z_1-z_2|^2\right\}$

   First compute each term.

3. Compute $|z-z_1|^2$ and $|z-z_2|^2$

   Since $z-z_1=(x-2)+i(y-3),$ we get $|z-z_1|^2=(x-2)^2+(y-3)^2.$

   Similarly, $z-z_2=(x-3)+i(y-4),$ so $|z-z_2|^2=(x-3)^2+(y-4)^2.$

4. Compute $|z_1-z_2|^2$

   $z_1-z_2=(2+3i)-(3+4i)=-1-i$

   Hence, $|z_1-z_2|^2=(-1)^2+(-1)^2=2.$

5. Substitute into the equation

   $[(x-2)^2+(y-3)^2]-[(x-3)^2+(y-4)^2]=2$

   Expand: $x^2-4x+4+y^2-6y+9-(x^2-6x+9+y^2-8y+16)=2$

   $x^2-4x+4+y^2-6y+9-x^2+6x-9-y^2+8y-16=2$

   Simplify: $2x+2y-12=2$

   $2x+2y=14$

   $x+y=7$

6. Interpret the locus

   The equation $x+y=7$ represents a straight line.

7. Find intercepts on coordinate axes

   • On the $x$-axis, $y=0$: $x=7$ So x-intercept $=7$.

   • On the $y$-axis, $x=0$: $y=7$ So y-intercept $=7$.

   Sum of intercepts: $7+7=14$

8. Check options

   • A: hyperbola with transverse axis 7  False
   • B: hyperbola with eccentricity 2  False
   • C: straight line with sum of intercepts $-18$  False
   • D: straight line with sum of intercepts $14$  True

Therefore, the set $S$ represents a straight line with the sum of its intercepts on the coordinate axes equal to $14$.