If the two (x – 1)2 + (y – 3)2 = r2 and x2 + y2 – 8x + 2y + 8 = 0 intersect in two distinct points, then


Correct option is

2 < r < 8

Centres of the given circles are C1(1, 3) and C2(4, –1), and their radii,r1 = r and r2 = 3.  

We know that the two circles touch, externally if C1C­2 = r1 + r2, and internally if C1C2 = |r1 – r2|.  

Thus the two circles will cut at two distinct points if C1C2 > |r1 – r2| and C1C2 < r1 + r2, i.e., if |r1 – r2| < C1C2 < r1 + r2




If two circles, each of radius 5 units, touch each other at (1, 2) and the equation of their common tangent is 4x + 3y = 10, then equation of the circle, a portion of which lies in all the quadrants is


A point moves such that the sum of the squares of its distances from the sides of a square of side unity is equal to 9. The locus of such a point is a circle


Radical centre of the three circles x2 + y2 = 9, x2 + y2 – 2x – 2y = 5, x2 + y2 + 4x + 6y = 19 lies on the line y = mx if m is equal to



The coordinates of the point on the circle x2 + y2 – 2x – 4y – 11 = 0 farthest from the origin are 


A circle passes through the origin O and cuts the axis at A(a, 0) and B(0,b). The reflection of O in the line AB is the point


The length of the longest ray drawn from the point (4, 3) to the circle x2y2 + 16x + 18y + 1 = 0 is equal to


1:- The chords in which the circle C cuts the members of the family S of circles through A and B are con-current at  


2:- Equations of the member of the family S which bisects the circumference of C is


3:- If O is the origin and P is the centre of C, then the difference of the squares of the lengths of the tangents from A and B to the circle is equal to


The distance between the chords of contact of the tangent to the circle x2 + y2 + 2gx + 2fy + c = 0 from the origin and the point (gf) is