A variable line passing through the origin intersects two given straight lines 2x + y = 4 and x + 3y = 6 at R and S respectively. A point P is taken on this variable line. Find the equation to the locus of the point P if?A

A) OP is the arithmetic mean of OR and OS.
B) OP is the geometric mean of OR and OS.
C) OP is the harmonic mean of OR and OS.

1 Answer
Feb 5, 2018

It is quite simple to give an answer in terms of parametric representation for all these curves - a different matter altogether to express them in "standard", i.e. x versus y form.

Explanation:

Let the variable line passing through the origin be y =tx where t represents the slope.
The point of intersection R of this variable line with the line2x+y=4 is given by
(4/{2+t},{4t}/{2+t})
(Substituting y=tx in 2x+y=4 gives (2+t)x=4 and so on ...)
Similarly, S, the intersection of y=tx with x+3y=6 is given by
(6/{1+3t},{6t}/{1+3t})

In (A) The point P is the midpoint of R and S and its coordinates are given by

x(t) =2/{2+t}+3/{1+3t}
y(t) = tx(t)
which is the parametric representation of the locus.

Of course, you can eliminate t by solving the equation for x(t) to obtain an expression for t in tems of x and substituting that in the equation for y(t) - but that is hardly likely to lead to any greater insight.

The same goes for the other two parts of the question.