LOCUS OF THE CENTRE OF A CIRCLE TOUCHING A GIVEN CIRCLE AND A GIVEN STRAIGHT LINE IS THE PARABOLA y^2=8x.THEN PROVE CENTRE OF THE GIVENCIRCLE IS(2,0). AND IF RADIUS OF THE GIVEN CIRCLE IS 2 UNITS, THEN GIVEN LINE MUST BE THE TANGENT TO THE CIRCLE.

Dear Abhijeet,

The question is quite simple.

The centre whose locus is y² = 8x is always at a constant distance with the given a cirle which is equal to the sum of their radius and also at a fixed distance with the straight line which is equal to the radius of the circle.

Therefore the ratio of both the distances is constant which goes with the basic defination of  parabola.

Here the point to which the circle is at constant distance is the centre of the given circle which can be found by the graph of parabola which comes out to be the focus of the parabola with co-ordinates (2, 0).

Hence the centre of the given circle is( 2,0 )

Thank you.