A dog wants to catch a cat. The dog follows the path whose equation is $y - x = 0$ while the cat follows the path whose equation is $x^{2} + y^{2} = 8$ . What are the coordinates of possible points of catching the cat?

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$(2, - 2)$ and $(- 2, - 2)$

$(- 2, 2)$ and $(2, 2)$

$(- 2, - 2)$ and $(- 2, 2)$

$(2, 2)$ and $(- 2, - 2)$

answer is B.

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Detailed Solution

Let catching point be $(x_{1}, y_{1}) .$ .

Then, $y_{1} - x_{1} = 0$ and $x_{1}^{2} + y_{1}^{2} = 8$ .

Therefore,

$\begin{array}{l} 2 {x_{1}}^{2} = 8 \\ {\Rightarrow x_{1}}^{2} = 4 \\ \Rightarrow x_{1} = \pm 2 \end{array}$

So, possible points are $(2, 2)$ and $(- 2, - 2)$ .

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