The question is as follows:
Find the equation of the straight line passing through $(-2,-7)$ and having intercept of length 3 units between straight lines $4x + 3y = 12$ and $4x + 3y = 3$.
I really didn't understand what they meant by "intercept of length 3 units". Hence, I couldn't proceed with the question.
I have a basic knowledge about the different forms of equations of a straight line, but none of them seemed to fit here.
Help will be appreciated. Thanks.
I agree it is difficult to interpret the question.
If you write the lines in the more useful $y=mx+c$ form you find
$$L_1:=y=-\frac43 x+3$$ and $$L_2:=y=-\frac{4}{3}x+1.$$
This means that the lines have the same slopes and so are parallel. Also the first cuts at $y=3$ and the second at $y=1$.
I think the "intercept of length three" means that the line we are looking for, $L$, cuts $L_1$ and $L_2$ at points $P_1=L\cap L_1$ and $P_2=L\cap L_2$ such that $$|P_1P_2|=3.$$
There are a number of ways of finishing this off. Synthetic geometry is probably easiest.
Consider the point $R$ on $L_1$ that is found vertically above $P_2$ and the triangle $\Delta (RP_1P_2)$.
We know the angle at $R$ (draw a picture and use the slope of $L_1$).
Using the Sine Rule we can thus find the angle at $P_1$ as $|P_2R|=2$ (why?).
Hence you know all the angles in particular the angle at $P_2$, $\alpha$.
The slope of $L$ is thus $$m=\tan\left(\frac{\pi}{2}-\alpha\right).$$