Find all the points on the y-axis aka (0, y) that are 6 units from the point (4, -3).
Seeking the steps to solve on my own.
Geometrically, open compasses to a distance 6 between the points. Set one point at (4, -3) and strike an arc of radius 6. The two points where that arc crosses the y-axis are the desired point. Of course, such an arc is a circle of radius 6 with center at (4, -3) which has equation $\displaystyle (x- 4)^2+ (y+ 3)^2= 36$. A point on the y-axis has x= 0 so the equation becomes $\displaystyle 16+ (y+ 3)^2= 36$. Then $\displaystyle (y- 3)^2= 36- 60= 20$. $\displaystyle y- 3= \pm\sqrt{20}= \pm 2\sqrt{5}$ and $\displaystyle y= 3\pm 2\sqrt{5}$.