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) 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

A) : `2x^2+7xy+3y^2-8x-9y=0.`

Explanation:

Let us name the given lines `l_1:2x+y=4, and l_2:x+3y=6.`

Let the variable line through the Origin be `l.`

Suppose that `l` makes an `/_ theta` with the `+ve` `X-` Axis.

Clearly, `l: y=xtantheta, or, ycostheta=xsintheta.`

Now, for any `P(X,Y) in l,` if the distance `OP=r,` then clearly,

`P(X,Y)=P(rcostheta,rsintheta).`

`:. X^2+Y^2=r^2, or, r=sqrt(x^2+y^2)........................(ast).`

Given that, `l nn l_1={R} rArr R in l, and R in l_1.`

`:."If "OR=r_1,"then, since "R in l, R(r_1costheta,r_1sintheta),`

and, since `R in l_1, 2(r_1costheta)+(r_1sintheta)=4.`

`:. r_1=4/(2costheta+sintheta).`

But, `(X,Y)=(rcostheta,rsintheta),`

`rArr r_1=4/{2(X/r)+Y/r}=(4r)/(2X+Y)............................(ast^1).`

`"Similarly, for "l nn l_2={S}, OS=r_2rArrr_2=6/(costheta+3sintheta).`

`rArr r_2=(6r)/(X+3Y).......................................................(ast^2).`

Recall that to find the Locus of `P(X,Y)` under the given conds.

A),B),C), we have to establish algebraic relation btwn. `X & Y,`

using A),B),C).

Part A) : `OP` is AM of `OR and OS.`

Under this cond., `2r=r_1+r_2.`

Then, by `(ast), (ast^1) and (ast^2),` we have,

`2r=(4r)/(2X+Y)+(6r)/(X+3Y), i.e.,`

`(2X+Y)(X+3Y)=2(X+3Y)+3(2X+Y), or,`

`2X^2+7XY+3Y^2-8X-9Y=0.`

Returning to traditional `(x,y),` the desired locus is given by,

`2x^2+7xy+3y^2-8x-9y=0.`

Parts B) and C) can be dealt with similarly.

Enjoy Maths.!