Statement 1 :The circles `x^2+y^2+2p x+r=0` and `x^2+y^2+2q y+r=0` touch if `1/(p^2)+1/(q^2)=1/edot` Statement 2 : Two centers `C_1a n dC_2` and radii `r_1a n dr_2,` respectively, touch each other if `|r_1+-r_2|=c_1c_2dot`

Statement I Circles x^(2)+y^(2)=4andx^(2)+y^(2)-6x+5=0 intersect each other at two distinct points Statement II Circles with centres C_(1),C_(2) and radii r_(1),r_(2) intersect at two distinct points if |C_(1)C_(2)|ltr_(1)+r_(2)

Let C_1 be the circle with center O_1(0,0) and radius 1 and C_2 be the circle with center O_2(t ,t^2+1),(t in R), and radius 2. Statement 1 : Circles C_1a n dC_2 always have at least one common tangent for any value of t Statement 2 : For the two circles O_1O_2geq|r_1-r_2|, where r_1a n dr_2 are their radii for any value of tdot

If radii of the smallest and the largest circle passing through ( sqrt( 3) ,sqrt(2)) and touching the circle x^(2) +y^(2) - 2 sqrt( 2)y -2=0 are r_(1) and r_(2) respectively, then find the mean of r_(1) and r_(2) .

Two circles centres A and B radii r_1 and r_2 respectively. (i) touch each other internally iff |r_1 - r_2| = AB . (ii) Intersect each other at two points iff |r_1 - r_2| ltAB lt r_1 r_2 . (iii) touch each other externally iff r_1 + r_2 = AB . (iv) are separated if AB gt r_1 + r_2 . Number of common tangents to the two circles in case (i), (ii), (iii) and (iv) are 1, 2, 3 and 4 respectively. circles x^2 + y^2 + 2ax + c^2 = 0 and x^2 + y^2 + 2by + c^2 = 0 touche each other if (A) 1/a^2 + 1/b^2 = 2/c^2 (B) 1/a^2 + 1/b^2 = 2/c^2 (C) 1/a^2 - 1/b^2 = 2/c^2 (D) 1/a^2 - 1/b^2 = 4/c^2

IF two circles of radii r_1 and r_2 (r_2gtr_1) touch internally , then the distance between their centres will be

The circle C_1 : x^2 + y^2 = 3, with centre at O, intersects the parabola x^2 = 2y at the point P in the first quadrant. Let the tangent to the circle C_1 at P touches other two circles C_2 and C_3 at R_2 and R_3, respectively. Suppose C_2 and C_3 have equal radii 2sqrt3 and centres at Q_2 and Q_3 respectively. If Q_2 and Q_3 lie on the y-axis, then (a) Q2Q3= 12(b)R2R3=4sqrt6(c)area of triangle OR2R3 is 6sqrt2 (d)area of triangle PQ2Q3 is= 4sqrt2

Two circles centres A and B radii r_1 and r_2 respectively. (i) touch each other internally iff |r_1 - r_2| = AB . (ii) Intersect each other at two points iff |r_1 - r_2| ltAB lt r_1 r_2 . (iii) touch each other externally iff r_1 + r_2 = AB . (iv) are separated if AB gt r_1 + r_2 . Number of common tangents to the two circles in case (i), (ii), (iii) and (iv) are 1, 2, 3 and 4 respectively. If circles (x-1)^2 + (y-3)^2 = r^2 and x^2 + y^2 - 8x + 2y + 8=0 intersect each other at two different points, then : (A) 1ltrlt5 (B) 5ltrlt8 (C) 2ltrlt8 (D) none of these