This is the question:
If a complex number z satisfies $\left|z+3\right|+\left|z-3\right|=10$, then the value of $\frac{60\left|z+3\right|}{\left|z+\overline{z}+\frac{50}{3}\right|}$ is?
I noticed that the first equation is that of the locus of an ellipse and the one in the denominator of the second expression is that of a line. Not sure of its significance in this question, just pointed it out if it helps to solve the question in a much shorter way.
I have solved the question assuming z=x+$i$y, forming quadratic equations and everything, but the whole process is too calculative and lengthy.
I am expected to solve this question in 3-5 mins during the test for which I am preparing. It would really help if anyone could think of a shorter method or point me in the right direction.
Note that $z + \bar{z} = -50/3$ is a directix of the ellipse whose corresponding focus is $-3$. We know that the ratio of the distance from the focus to that of from the corresponding directix is the eccentricity of the ellipse. In the given ellipse, the length of major axis is $10$ and distance between foci is $6$ giving an eccentricity of $0.6$. Thus the required answer is $60 \cdot 0.6 = 36$.
The usual trick in such questions is that required quantity that you have to calculate is somehow fundamentally related to the locus. Thus, identifying the locus as an ellipse, like you did, is the correct direction to proceed in.