Paragraph: Let C₁: x^2+y^2=r^2 and C₂:(x-p)^2+(y-q)^2=r^2 be 2 circles with radius r(r>0) and have n points of intersection, (x_i, y_i ) for i 1,2, , n . If C₁ and C₂ are orthogonal at all points of intersection, then: Question: As we move the centre of C₂ along the curve q=a for some constant a 0 , then d r d p is equal to:

Step-by-step solution

aligned & (x-p)^2+(y-q)^2=x^2+y^2=r^2 & p^2+q^2=2 p x+2 q y ...(1) aligned For the circle to intersect orthogonally aligned & . . d y d x |^ C₁ d y d x |^ C₂ =-1 & ( -x y ) ( -(x-p) y-q )=-1 & x^2-p x=q y-y^2 x^2+y^2-p x-qy=0 ...(2) aligned From eqns. (1) and (2) p^2+q^2=2 (x^2+y^2 )=2 r^2...(3) Now, q=a differential 2 p+0=4 r d r d p d r d p = p 2 r Now, p+b q=0 put p=-b q in eqn. (3) aligned & (b^2+1 ) q^2=2 r^2 & (b^2+1 ) 2 q=4 r d r d q aligned d r d q = (b^2+1 ) q 2 r