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/rdot` 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`

If S_1=x^2+y^2+2g_1x+2f_1y+c_1=0 and S_2=x^2+y^2+2g_2x +2f_2y+c_2=0 are two circles with radii r_1 and r_2 respectively, show that the points at which the circles subtend equal angles lie on the circle S_1/r_1^2=S_2/r_2^2

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

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

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. Number of common tangents to the circles x^2 + y^2 - 6x = 0 and x^2 + y^2 + 2x = 0 is (A) 1 (B) 2 (C) 3 (D) 4

C is the center of the hyperbola (x^2)/(a^2)-(y^2)/(b^2)=1 The tangent at any point P on this hyperbola meet the straight lines b x-a y=0 and b x+a y=0 at points Qa n dR , respectively. Then prove that C QdotC R=a^2+b^2dot

If the two circles (x+1)^2+(y-3)^2=r^2 and x^2+y^2-8x+2y+8=0 intersect at two distinct point,then (A) r > 2 (B) 2 < r < 8 (C) r < 2 (D) r=2

The circle C1 : x^2 + y^2 = 3 , with center at O, intersects the parabola x^2 = 2y at the point P in the first quadrant. Let the tangent to the circle C1 at P touches other two circles C2 and C3 at R2 and R3, respectively. Suppose C2 and C3 have equal radii 2sqrt(3) and centers Q2 and Q3, respectively.If Q_2 and Q_3 lies on the y-axis, then

There are two perpendicular lines, one touches to the circle x^(2) + y^(2) = r_(1)^(2) and other touches to the circle x^(2) + y^(2) = r_(2)^(2) if the locus of the point of intersection of these tangents is x^(2) + y^(2) = 9 , then the value of r_(1)^(2) + r_(2)^(2) is.