Question

A cat wants to catch a rat. The cat follows the path whose equation is$x + y = 0$. But the rat follows the path whose equation is${x^2} + {y^2} = 4$. The coordinates of possible points of catching the rat are:
(a) $\left( {\sqrt 2 } \right.,\left. {\sqrt 2 } \right)$
(b) $\left( { - \sqrt 2 } \right.,\left. {\sqrt 2 } \right)$
(c) $\left( {\sqrt 2 } \right.,\left. {\sqrt 3 } \right)$
(d) $\left( 0 \right.,\left. 0 \right)$

Answer 1

n this solution, we are going to solve the two equations given, by substitution method. The coordinates that we get on solving the two equations, are the possible points of catching the rat.

Complete Step by Step Answer: Given:
The equation of path followed by cat: $x + y = 0\;\;.......(1)$
The equation of path followed by rat: ${x^2} + {y^2} = 4\;\;......(2)$
The coordinates of possible points of catching the rat, can be obtained by solving the above two equations for $x$ and $y$
From equation (1), we have:
$x + y = 0 \\\ or\;x = - y\;\;......(3) \\\$
Substituting the value of $x$ from equation (3) in equation (2), we get,
${\left( { - y} \right)^2} + {y^2} = 4 \\\ {y^2} + {y^2} = 4 \\\ 2{y^2} = 4 \\\ y = \pm \sqrt 2 \;\;\;.......(4) \\\$
So, from equation (4), we have two values of $y$ as:
$y = \sqrt 2$ and $y = - \sqrt 2$
From Equation (3): $x = - y\;\;......(3)$
Substituting these values of $y$ in equation (3), we get corresponding values of$x$:
For $y = \sqrt 2$ , $x = - \sqrt 2$ and,
For$y = - \sqrt 2$, $x = \sqrt 2$
So, the coordinates of possible points of catching the rat are:
$\left( { - \sqrt 2 } \right.,\left. {\sqrt 2 } \right)$ and $\left( {\sqrt 2 } \right., - \left. {\sqrt 2 } \right)$
As, the coordinate $\left( {\sqrt 2 } \right.,\left. { - \sqrt 2 } \right)$ is not given in the options, so, coordinate $\left( { - \sqrt 2 } \right.,\left. {\sqrt 2 } \right)$ will be the possible point of catching the rat.
Therefore, option (b) is the correct answer.
Note: Alternate method:
We can also find the correct answer by substituting the values of $x$ and $y$ from the given options, in the two equations $x + y = 0$and${x^2} + {y^2} = 4$.
The coordinates that satisfy both the equations will be the correct answer. On substituting, we can see that only the coordinate $\left( { - \sqrt 2 } \right.,\left. {\sqrt 2 } \right)$ satisfies both the equations.
Therefore, option (b) $\left( { - \sqrt 2 } \right.,\left. {\sqrt 2 } \right)$ is the possible coordinate point of catching the rat.