If `S=0 and S_1=0` are the two circles with radii `a and a_1` respectively. Show that the circle `S/a +- S_1/a_1 = 0` intersect at right angles.

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

If the equations of two circles, whose radii are r and R respectively, be S = 0 and S' = 0, then prove that the circles S/r+-(S')/R=0 will intersect orthogonally

The length of the common chord of two circles of radii r_1 and r_2 which intersect at right angles is

If S = 0, S' =0 are two touching circles then angle between their radical axis and the common tangent at the at their points of contanct is

Let A B C be a triangle right-angled at Aa n dS be its circumcircle. Let S_1 be the circle touching the lines A B and A C and the circle S internally. Further, let S_2 be the circle touching the lines A B and A C produced and the circle S externally. If r_1 and r_2 are the radii of the circles S_1 and S_2 , respectively, show that r_1r_2=4 area ( A B C)dot

A mixture consists of two radioactive materials A_1 and A_2 with half-lives of 20 s and 10 s respectively. Initially the mixture has 40 g of A_1 of 160 g of A_2 . After what time of amount of the two in the mixture will become equal ?

Assertion: Distance between two coherent sources S_1 and S_2 is 4lambda . A large circle is drawn around these sources with centre of circle lying at centre of S_1 and S_2 . There are total 16 maxima on this circle. Reason: Total number of minimas on this circle are less, compared to total number of maximas.

The equation of a circle S_1 is x^2 + y^2 = 1 . The orthogonal tangents to S_1 meet at another circle S_2 and the orthogonal tangents to S_2 meet at the third circle S_3 . Then (A) radius of S_2 and S_3 are in ratio 1 : sqrt(2) (B) radius of S_2 and S_1 are in ratio 1 : 2 (C) the circles S_1, S_2 and S_3 are concentric (D) none of these

A circle arc of radius 1 subtends an angle of x radians as shown in figure. The centre of the circle is O and the point C is the intersection of two tangent lines at A and B. Let T(x) be the area of DeltaABC and S(x) be the area of shaded region. lim_(xto0)(S(x))/x is

A circle x^2 +y^2 + 4x-2sqrt2 y + c = 0 is the director circle of circle S_1 and S_1 is the director circle of circle S_2 , and so on. If the sum of radii of all these circles is 2, then find the value of c.