The minimum distance between the curves y2=4x and x2+y2−12x+31=0 is

Centre and radius of the given circle are P(6,0) and 5 , respectively.Now, the shortest distance always occurs along the common normal. Differentiating y2=4x with respect to x we get,dydx=2yThen the slope of nonnal at point Ay12/4,y1 is −y1/2 Also, from definition, the slope of AP is given by y1−0y12/4−6=−y12 i.e., y1=0 or y1=±4 Hence, the points are O(0,0),A(4,4), and C(4,−4) .  The shortest distance is AP−5=20−5=5 .