Find where the circle (x - 0.5334)^2 + (y - 0.5334)^2 = 0.3475^2 meets the line sqrt(3)/3 * x - y = 0.

Step-by-step solution

1. 1

   Rewrite the line as $y$ in terms of $x$.

   $y=\frac{\sqrt3}{3}x=\frac{x}{\sqrt3}\approx 0.5773503\,x$

   The line passes through the origin at an angle of $30^\circ$, since its slope is $\tan 30^\circ$.

2. 2

   Substitute into the circle equation. Replacing $y$ turns a two-variable system into a single quadratic:

   $(x-0.5334)^2+\left(\frac{x}{\sqrt3}-0.5334\right)^2=0.3475^2$

   Expanding and collecting powers of $x$ gives $Ax^2+Bx+C=0$ with

   $A=1+\frac13=\frac43,\qquad B=-2(0.5334)-\frac{2(0.5334)}{\sqrt3}=-1.6827173,\qquad C=2(0.5334)^2-0.3475^2=0.4482749$

3. 3

   Test the discriminant before solving.

   $\Delta=B^2-4AC=2.8315371-2.3907990=0.4407381>0$

   A positive discriminant means the line genuinely cuts the circle twice; zero would mean tangency and negative would mean the line misses it entirely. Checking this first tells you how many answers to expect.

4. 4

   Solve the quadratic.

   $x=\frac{1.6827173\pm\sqrt{0.4407381}}{2(4/3)}=\frac{1.6827173\pm 0.6638811}{2.6666667}$

   $x_1=0.879974,\qquad x_2=0.382064$

5. 5

   Recover the $y$-coordinates from the line.

   $y_1=0.5773503(0.879974)=0.508053,\qquad y_2=0.5773503(0.382064)=0.220585$

   so the intersection points are $(0.8800,\,0.5081)$ and $(0.3821,\,0.2206)$.

6. 6

   Verify both points lie on the circle. Substituting back, $(0.879974-0.5334)^2+(0.508053-0.5334)^2=0.120756=0.3475^2\;\checkmark$, and the same holds for the second point to twelve decimal places. Note that rounding the first point to $(0.8806,\,0.5084)$ would put it off the circle by about $4\times 10^{-4}$.