Question

The equation of the circle passing through (2,0) and (0,4) and having the minimum radius is

Answer 1

Answer:

Equation if line joining the two given points is 3y-4x=0 which is satisfied th centre if the circle hence one of the diameters of the given circle will lie on the line joining the origin and (12,16) . Hence the shortest path's distance will be Distance between origin and given point - length of the line inside the circle(diameter) + half the circumference of the circle (as this will be the path used instead of the diameter)

Required distance = 20 - 10 + 5*pi = 25.71 units.

Edit : above answer is what one would get at first glance but distance comes out to be different with a correct and different approach.

The required path is OABO'. OA and OB can be found by dropping normal from centre to the tangents, forming a right angle triangle and thereafter applying the Pythagoras's theorem .

OA = O'B = 5√3

In order to find the arc length AB we need the angle theta . Using those right angle triangles again the other angle phi is found to be 60 degrees, this theta becomes 180 - 2(phi) = 60 degrees.

So now arc length = (theta/360)*2*pi*radius = (5/3)*pi

So the total distance is = 5√3 + (5/3)*pi+ 5√3 = 22.55 units which turns out to be lesser than that found by first approach