a straight line makes an intercept on the y axis twice as long as that on the x axis and is at a unit distance from the origin.determine its equation

Let us first concentrate on 1st quadrant.

Let the line cut the X-axis at A (a,0) and the Y-axis at B(0,2a).

O being the origin, what is the area of triangle AOB = (1/2)*OA*OB = a^2…. (i)

Now,what is the length of AB?

By Pythagoras, it is sqrt(a^2+4a^2) = sqrt(5)a.

The line AB is at a distance 1 from O. Drop a perpendicular OH from O on AB so that OH=1.

In this way, area of triangle AOB = (1/2)*OH*AB = (1/2)*1*(sqrt(5))a…. (ii)

Equating (i) and (ii):

a=sqrt(5)/2.

Now, you can easily see that the eqn. of the line is x/a+y/2a=1.

You have to do this on all the 4 quadrants to get the correct signs of the eqns. i.e. take all possible positive / negative x / y combinations to get all possible lines.