Question

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.

Answer 1

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 line`2x+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.