Please help me with the process of finding the equation of a circle in the first quadrant that is tangent to the x-axis, y-axis, and a line of the form ax+by+c=0. Thanks.
A circle is tangent to the x-axis iff , is tangent to the y-axis iff , and since the circle's center is in the first quadrant this means that the circle's equation is , and center at
Finally, the distance from a given point to a given line is given by . Do some maths now.
Tonio
if the circle is only in the first quadrant, here is another way to do it.
See the attachment...
the triangle contains pairs of identical right-angled triangles,
hence...
where c is the length of the hypotenuse, the 3rd side of the triangle.
Then it is straightforward to find r.
Tonio's method finds both solutions.
the circle of r=6 is also tangent to all 3 lines.
In addition to the previous posts I've done a geometrical construction: Yo'll find the two centers as Points of intersection of two of the three angle bisectors.
The colour of the bisector correspond to the colour of the circle. All three angle bisectors form a right triangle.