A doctor is located at X and his hospital is located at Y. There is no straight lane from the doctor's house to the hospital. So, the doctor first goes to A which is 600 m due east of X, then goes to B which is 450 m due south of A, then to C which is 120 m due west of B and finally reaches the hospital at Y which is 90 m due north of C. With respect to the hospital the correct direction of the doctors house is

The correct option is north-west.

To find the direction of the doctor's house (X) with respect to the hospital (Y), let us trace the path of the doctor step-by-step using a standard Cartesian coordinate system where East is along the positive x-axis, West is along the negative x-axis, North is along the positive y-axis, and South is along the negative y-axis.

Let the doctor's starting position at house X be the origin:
$\text{X}=(0,0)$

Step 1: The doctor goes to point A, which is 600 m due east of X.
$\text{A}=(600,0)$

Step 2: From A, he goes to point B, which is 450 m due south of A.
$\text{B}=(600,0-450)=(600,-450)$

Step 3: From B, he goes to point C, which is 120 m due west of B.
$\text{C}=(600-120,-450)=(480,-450)$

Step 4: Finally, from C, he reaches the hospital at Y, which is 90 m due north of C.
$\text{Y}=(480,-450+90)=(480,-360)$

Now, we need to find the direction of the doctor's house X relative to the hospital Y.

The vector representing the position of X from Y is:
$\overrightarrow{\text{Y}\text{X}}=\text{X}-\text{Y}=(0-480,0-\left(-360\right))=(-480,360)$

Since the x-coordinate of house X relative to hospital Y is negative (-480 m), house X lies to the west of Y.
Since the y-coordinate of house X relative to hospital Y is positive (+360 m), house X lies to the north of Y.

Combining these two components, with respect to the hospital, the doctor's house is in the north-west direction.