Let r 1 and r 2 be the radii of the largest and smallest circles, respectively, which pass through the point ( - 4 , 1 ) and having their centres on the circumference of the circle x 2 + y 2 + 2 x + 4

Question

Let r 1 and r 2 be the radii of the largest and smallest circles, respectively, which pass through the point ( - 4 , 1 ) and having their centres on the circumference of the circle x 2 + y 2 + 2 x + 4 y - 4 = 0 . If r 1 r 2 = a + b 2 , then a + b is equal to:

TestHub · Accepted Answer

We have, x 2 + y 2 + 2 x + 4 y - 4 = 0 ⇒ x 2 + 2 x + 1 + y 2 + 4 y + 4 = 9 ⇒ x + 1 2 + y + 2 2 = 3 2 Centre ≡ - 1 , - 2 C P = - 4 + 1 2 + 1 + 2 2 = 3 2 Centre of smallest circle is A Centre of largest circle is B . r 2 = | C P - C A | = 3 2 - 3 r 1 = C P + C B = 3 2 + 3 Hence, r 1 r 2 = 3 2 + 3 3 2 - 3 = ( 3 2 + 3 ) 2 9 ⇒ r 1 r 2 = ( 2 + 1 ) 2 = 3 + 2 2 Hence, a = 3 , b = 2 a + b = 5

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