Question

If the circles `x^(2)+y^(2) +5Kx+2y + K=0 and2(x^(2)+y^(2))+2Kx +3y-1=0, (KinR)`, intersect at the points P and Q, then the line 4x + 5y - K = 0 passes through P and Q, for `x^(2)+y^(2)-16x-20y+164=r^(2)` and `(x-4)^(2)+(y-7)^(2) = 36` intersect at two distinct points, then
A. `0ltrlt1`
B. `r gt11`
C. `1 lt r lt11`
D. r= 11

Answer 1

Correct Answer - C
Circle I is `x^(2)+y^(2)-16x-20y+164=r^(2)`
`(x-8)^(2)+(y-10)^(2)=r^(2)`
`rArr C_(1)(8,10) " is the centre of Istcircle and " r_(1)=r " is its radius Circle II is "(x-4)^(2)+(y-7)^(2)=36`
`rArr C_(2)(4,7) " is the centre of 2nd circle and " r_(2)=6 " is its radius "`.
Two circles intersect if `|r_(1)-r_(2)|ltC_(1)C_(2)ltr_(1)+r_(2)`
`rArr|r-6|ltsqrt((8-4)^(2)+(10-7)^(2)) ltr+6`
`rArr|r-6|ltsqrt(16+9)ltr+6`
`rArr |r-6|lt5ltr+6`
`" Now as ", 5ltr+6` always, we have to solve only
`|r-6|lt5rArr-5ltr-6lt5`
`rArr6-5ltrlt5+6rArr1ltrlt11`