Thread: Cat and mouse

Okay so .....

A mouse is running a donut-shaped track with equation:

(X-4)^2 + (Y-3)^2 = 4

At the same time a cat is running a coplanar intersecting opaque track with equation:

[(X-4)^2] ÷ 9 + [(Y-3)^2] ÷ 1 = 1

What are the four possible coordinates of the intersections (simultaneous solutions) where the mouse might catch the cat?

Thanks so much if you can help me out

Originally Posted by gdunne18

]A mouse is running a donut-shaped track with equation:
(X-4)^2 + (Y-3)^2 = 4
At the same time a cat is running a coplanar intersecting opaque track with equation:
(X-4)^2/ 9 + (Y-3)^2/ 1 = 1
What are the four possible coordinates of the intersections (simultaneous solutions) where the mouse might catch the cat?...

Hello,

there are a lot of different ways to get the 4 points. In this case it would be the best to do the calculations like this:

(X-4)^2 + (Y-3)^2 = 4 ===> (Y-3)^2 = 4 - (X-4)^2

(X-4)^2/ 9 + (Y-3)^2/ 1 = 1 ===> (Y-3)^2 = 1 - (X-4)^2/ 9 Therefore:

4 - (X-4)^2 = 1 - (X-4)^2/ 9 ===> 3 = (X-4)^2 - (X-4)^2/ 9

3 = (8/9)*(X-4)^2 ===> (27/8) = (X-4)^2 (******)

x = 4 ± (3/4)*√(6)

Use this (*****) result to calculate y:

(Y-3)^2 = 4 - (X-4)^2 ===> (Y-3)^2 = 4 - (27/8) = (5/8)

y = 3 ± (1/4)*√(10)

The 4 points are:
(4 + (3/4)*√(6), 3 + (1/4)*√(10)); (4 + (3/4)*√(6), 3 - (1/4)*√(10)); (4 - (3/4)*√(6), 3 - (1/4)*√(10)); (4 - (3/4)*√(6), 3 - (1/4)*√(10))

EB

Hello, gdunne18!

A mouse is running a circular track with equation: .(x - 4)² + (y - 3)² .= .4

At the same time a cat runs a track with equation: .(x - 4)²/9 + (y - 3)² .= .1

What are the four possible coordinates of the intersections
where the mouse might catch the cat?

We have: . (x - 4)² + .(y - 3)² . = .4 . [1]
. . . .and: . (x - 4)² + 9(y - 3)² -= .9 . [2]

Subtract [1] from [2]: .8(y - 3)² .= .5
. . . . . . . . . . . . . . . . .__
Solve for y: .y .= .3 ± √10/4
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ._
Substitute into [1] and solve for x: .x .= .4 ± 3√6/4

. . . . . . . . . . . . . . . . . . . . . . _ . . . . . . __
The intersections are: . (4 ± 3√6/4, 3 ± √10/4)

thanks so much you guys big help thanks again