A weird locus question. (2 Viewers)

[Tsylana]

This is from Fitzpatrick 2U

A ladder that is 6m long rests with one end on the horizontal ground, and the other end against a vertical wall. Considering the ground and the wall as the X and Y axes respectively find the midpont of the ladder.

Well... I got to the answer... I had a few problems proving it with different methods though and some of the things I tried to do just plain didnt seem to work so... yeah.. discuss . I want to see what you people think... XD.


The answers

x^2 + y^2 = 9

0 < x < 3
0 < y < 3

edit edit...

i actually have a problem doing this question, its just really confusing for me to picture what im actually suppose to do ".

A is a point where the crcle x^2 + y^2 = 16 cuts the X axis find the locus of the midpoint of all chords on this circle that contain point A

Q 13 from 10 ( c ) 2u fitzpatrick if anyones wondering

 

[3unitz]

This is from Fitzpatrick 2U

A ladder that is 6m long rests with one end on the horizontal ground, and the other end against a vertical wall. Considering the ground and the wall as the X and Y axes respectively find the midpont of the ladder.

Well... I got to the answer... I had a few problems proving it with different methods though and some of the things I tried to do just plain didnt seem to work so... yeah.. discuss . I want to see what you people think... XD.


The answers

x^2 + y^2 = 9

0 < x < 3
0 < y < 3

let x₀ denote the horizontal distance from the wall to the base of the ladder, and y₀ denote the vertical distance from the ground to the top of the ladder.

length of the ladder is 6m which implies:
x₀² + y₀² = 6² -----(1)

the midpoint has parameters:
x = x₀/2 => x₀ = 2x -----(2)
y = y₀/2 => y₀ = 2y -----(3)

sub (2) and (3) into (1):
(2x)² + (2y)² = 6²
x² + y² = 9

 

[3unitz]

i actually have a problem doing this question, its just really confusing for me to picture what im actually suppose to do ".

A is a point where the crcle x^2 + y^2 = 16 cuts the X axis find the locus of the midpoint of all chords on this circle that contain point A

Q 13 from 10 ( c ) 2u fitzpatrick if anyones wondering

let (x₀, y₀) denote a point which lies on the circle.
i.e. x₀^2 + y₀^2 = 16 -----(1)

the circle cuts the x axis at A (+4, 0)

midpoint has parameters:
x = (x₀ + 4) /2 => x₀ = 2x + 4 -----(2)
y = y₀ /2 => y₀ = 2y -----(3)

sub (2) and (3) into (1):
(2x + 4)^2 + (2y)^2 = 16

 

[Tsylana]

I figured out the first one I was just more or less wondering where the restriction came about "

 

[lolokay]

let (x₀, y₀) denote a point which lies on the circle.
i.e. x₀^2 + y₀^2 = 16 -----(1)

the circle cuts the x axis at A (+4, 0)

midpoint has parameters:
x = (x₀ + 4) /2 => x₀ = 2x + 4 -----(2)
y = y₀ /2 => y₀ = 2y -----(3)

sub (2) and (3) into (1):
(2x + 4)^2 + (2y)^2 = 16

that simplifies to (x + 2)^2 + y^2 = 4
just half the radius of the old circle to find the midpoints, and change x and y in accord with the new centre

I figured out the first one I was just more or less wondering where the restriction came about "

this one?
0 < x < 3
0 < y < 3

the y bit means that the since the ladder will only be above the ground, y >= 0. I'm not sure how you would know the x bit though, I would think -3<x<0 would be acceptable too. I don't know if you would need to add in the 3 either, since that's pretty much given by the locus