An Equation of Circles and Squares

Note the the first equation given is that of a 'tilted square' centred at the origin, and the second equation given is that of a circle with radius $\text{a}$ , centred also at the origin, where the origin is $\displaystyle \text{(0,0)}$ .

The two conditions where there are exactly four intersections are:

• Circle is inscribed in the square.

• Circle passes through the vertices of the square.

In case I , the radius of the circle is $\displaystyle a = \frac{1}{\sqrt{2}}$ . This is because the circle passes through the point $\displaystyle (\frac{1}{2},\frac{1}{2})$ (This is self-evident, as the circle is inscribed inside the square)

In case II , the radius of the circle is $\displaystyle a= 1$ . This is because the circle passes through the point $\displaystyle (1,0)$ .

Note that we are done and the required range of a is $\displaystyle \boxed{ \frac{1}{\sqrt{2}} <\text{ a }< 1 }$ .

This is sufficient as:

• if $\displaystyle a<\frac{1}{\sqrt{2}}$ , there will be no intersections between the circle and this 'tilted square' and

• if $\displaystyle a>1$ , the same will be the case.

Cheers!