Question

A cat wants to catch a rat. The cat follows the path whose equation is `x+y=0`. But rat follows the path whose equation is `x^(2)+y^(2)=4`. The coordinates of possible points of catching the rat are
A. `(sqrt(2), sqrt(2))`
B. `(-sqrt(2),sqrt(2))`
C. `(sqrt(2),sqrt(3))`
D. `(0,0)`

Answer 1

(b) The paths of rat and cat are shown in figure. The possible points of catching the rat by the cat are `A(x_(1),y_(1))` and `B(x_(2), y_(2))`. From figure

`OA=OB=` radius of circle `=2` unit
`:. x_(1)=-OA sin 45^(@)=-2/(sqrt(2))=2sqrt(2)`
and `y_(1)=OA sin 45^(@)=sqrt(2)`
Similarly for point `B, x_(2)=sqrt(2), y_(2)=-sqrt(2)`
Alternate
Let the catching point of rat by cat is `P(x_(1),y_(1))`.
The coordinates of point `P` satisfy the both equations.
`:. x_(1)+y_(1)=0`
`:. y_(1)=-x_(1)`..............(i)
From rate, `x_(1)^(2)+y_(1)^(2)=4`
or `x_(1)^(2)+(-x_(1))^(2)=4`
`:. 2x_(1)^(2)=4`
`:. x_(1)=+-sqrt(2)`
The corresponding values of `y_(1)=-+sqrt(2)`
`:.` The possible coordinates of point `P` are `(sqrt(2),-sqrt(2)` and `(-sqrt(2),sqrt(2))`