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This problem trains for: SAT-II, AMC-10.

The lines:

equation

equation

are intersected by the line:

equation

What is a value of the parameter k for the triangle formed by the intersections of these three lines to have area 36 square units?

From the figure we can see that there are two possible positions of the vertical lines for which the area of the triangle can be made to have a certain value:

figure

Both lines AC and DF are going to give valid solutions.

We cannot get both solutions from a single equation, because the parametrizations of the two cases are different!

Therefore, we have two options: either to solve for each case in part or to translate the figure left so as to make it symmetrical. Since the lines intersect at (1,0), we have to translate left by one unit:

figure

In this much simpler case, the triangle ODF is a right isosceles triangle (since the oblique lines have slopes that are negative reciprocals of one another). The legs OD and OF have length:

equation

and the area of the triangle ODF is:

equation

equation

equation

We obtained two values for p, which accounts for the two possible positions of the vertical lines.

Now we have to translate the figure back to its original position. The two values for k are:

equation

equation