Question

find the equation of the circle which has radius 5 units and which is tangent to the line 3x-4y-16=0 at the point (4,1)​

Answer 1

As stated there are infinite answers to this question, however assuming that you mean is tangent to instead of touches this will still give you two possible answers.

First you find the center by moving 5 units from that point in a direction that is perpendicular to the line given. This provides two centers for two possible answers. The x,y coords of center are (x0,y0)

Then you plug in the standard notation for a cartesian circle equation:

(x−x0)2+(y−y0)2=r2

So lets find out what direction is perpendicular to this line by converting it to the form of y=mx+b and then taking the orthogonal slope of -1/m

3x+4y=11

4y=−3x+11

y=−3x+114

The slope is -3/4 so the orthogonal slope is 4/3.

Moving along this direction from point (1,2) will give us our two possible circle centers as either (4,6) or (-2,-2)

Equation for this circle is either:

(x+2)2+(y+2)2=52

or

(x−4)2+(y−6)2=52