The equation of a circle is x^2+y^2=4.
The centre of the smallest circle touching this circle and the line
The given line x+y=52, has a slope of -1. It makes an angle 45 with x axis, on the given circle side.
Draw a perpendicular to this line from origin, and what we get is a right angled triangle, whose hypotenuse is the 52.
The perpendicular distance from the origin i.e. centre of the given circle onto the line, x+y = 52,
is, 52 sin45 = 5 and it's a line with slope 1, since it's perpendicular to the line of slope -1.
But this perpendicular distance will be,
Radius of the given circle + Diameter of the smallest circle required,
2 + d = 5
d = 3
And the smaller circle lies on this line, i.e. y = x,
The distance of the centre of smaller circle from origin, is 2+1.5 = 3.5.
Hence it's x coordinate, 3.5 sin 45 = 72/4.
Therefore, the coordinates of the smaller circle, (72/4, 72/4).