Quadratic EquationHard
Question
How many solutions does the system of equations $|x| + |y| = 1,x^{2} + y^{2} = a^{2}$ possess depending on ' $a$ '?
Options
A.if $|a| < \frac{1}{\sqrt{2}}$
B.0 if $|a| > 1$
C.4 if $|a| = \frac{1}{\sqrt{2}},1$
D.8 if $\frac{1}{\sqrt{2}} < |a| < 1$
Solution
$|a| = sin\frac{\pi}{4} = \frac{1}{\sqrt{2}}$
Number of solutions
$$= \left\{ \begin{matrix} 4 & \text{~if~} & a = \left\{ \frac{1}{\sqrt{2}}, - \frac{1}{\sqrt{2}}, - 1,1 \right\} \\ 8 & \text{~if~} & a \in \left( - 1, - \frac{1}{\sqrt{2}} \right) \cup \left( \frac{1}{\sqrt{2}},1 \right) \\ 0 & \text{~if~} & a \in ( - \infty, - 1) \cup \left( - \frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}} \right) \cup (1,\infty) \end{matrix} \right.\ $$
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