Sliding Laddder PQ, w/ R between, RQ = 2*RP: find locus

[Yuzu613]

5. A ladder PQ, 5 meters long, is resting against a wall. (P is against the wall; Q is on the floor, away from the wall.) Assume that the floor and the wall represent the x- and y-axis, respectively, and that R is a point on PQ such that RQ is twice RP. Find:

(a) the locus of point R when ladder PQ slides down the wall, and

(b) the coordinates of point R when it touches the floor.

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As the picture shown, I got no clues how to solve it, can someone solve it with details step-by-step?

Here are my steps,

First I get the coordinate of R by using the formula and I get R(a/3 , 2b/3)

Since the ladder is 5m long, so x²+y²=25, following by x²=25-y² and y²=25-x², a²=25-y² b²=25-x²

Then I calculated the distance from the point R to the origin, O(0,0) and I get sqrt( (a/3-0)²+(2b/3-0)² )

After that I don't know how to proceed anymore...

 

[Deleted member 4993]

5. A ladder PQ, 5 meters long, is resting against a wall. (P is against the wall; Q is on the floor, away from the wall.) Assume that the floor and the wall represent the x- and y-axis, respectively, and that R is a point on PQ such that RQ is twice RP. Find:

(a) the locus of point R when ladder PQ slides down the wall, and

(b) the coordinates of point R when it touches the floor.

---

As the picture shown, I got no clues how to solve it, can someone solve it with details step-by-step?

What are your thoughts?

Please share your work with us ...even if you know it is wrong.

If you are stuck at the beginning tell us and we'll start with the definitions.

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[Yuzu613]

Solved

Well somehow I solved the question,

First lets assume the coordinate for R is R(x,y)

By using the formula (nx1+mx2/n+m , ny1+my2/n+m), we get the coordinate of R which is R(a/3,2b/3)

Since R(x,y)=R(a/3,2b/3), so x=a/3 and we get 3x=a

The ladder is 5m long and divided to 3 segments, so each segment is 5/3 long, so the distance between R and Q is 10/3.

By using the distance formula, 10/3=sqrt(x-a/3)²+(y-0)²

and we will get the locus of the sliding ladder which is 36x²+9y²​-100=0