Answer to Question #102780 in Analytic Geometry for eLLA

Question

How many lines on the coordinate plane that has a distance 1 to the origin O(0,0) and distance 2 to the point A (3,0)?

Answer 1

All points on the coordinate plane that has a distance 1 to the origin O(0,0) are situated on the circle, centred at (0,0) with radius 1.

Analogically, all points on the coordinate plane that has a distance 2 to the point A (3,0) are situated on the circle, centred at A(3,0) with radius 2.

Drawing the circles centred at (0,0) and (3,0) with radii 1 and 2 respectively, you can easily see that they osculate (touch) at the point (1,0).

Since these two circles osculate and none of them is laying inside the other, there is a line that touches both circles at the point of osculation, and two other lines which touch these two circles (one for y>0 and the other for y<0). Therefore, there would be 3 lines.

See the picture