Q.
AB is any chord of the circle x2+y2−6x−8y−11=0, which subtends 90∘ at (1,2) . If locus of mid-point of AB is a circle x2+y2−2ax−2by−c=0, then value of (a+b+c) is
Let (h,k) be the mid-point of chord AB . (x−h)2+(y−k)2=(h−1)2+(k−2)2 ⇒x2+y2−2hx−2ky+2h+4k−5=0...(1)
Given equation of circle is x2+y2−6x−8y−11=0...(2) ⇒ Equation of chord is (1),(2) (h−3)x+(k−4)y−h−2h−3=0...(3)
Equation of chord w.r.t. midpoint is T=S1 xh+ky−3(x+h)−4(y+k)−11 =h2+k2−6h−8k−11...(4)
Comparing (3) and (4) 3h+4k−h2−k2=−h−2k−3 ⇒h2+k2−4h−6k−3=0 ∴a=2;b=3;c=3 ∴a+b+c=8