Question

Let S1=x^2+y^2-2x-6y+9=0 and S2 = x^2+y^2+2y-8=0 Be two circles with center C1, C2 and radius R1 and R2 respectively then C1C2+R1+R2 equals (A) 4+√17 (B) 4+√5 (C) 5+√5 (D) 5+√17

Answer 1

Answer: hey there,

Step-by-step explanation:If k = –1, we get equation of common chord i.e. straight line instead of circle.  

Let S1 ≡ x2 + y2 + 2g1x + 2f1y + c1 = 0  

S2 ≡ x2 + y2 + 2g2x + 2f2y + c2 = 0

Since, point lies on both the circles,

⇒ xA2 + yA2 + 2g1xA + 2f1yA + c1 = 0

So, xA2 + yA2 + 2g2xA + 2f2yA + c2 = 0

⇒ xA2 + yA2 + 2g1xA + 2f1yA + c1 + λ (xA2 + yA2  + 2g1xA + 2f1yA + c1) = 0

⇒ Point A(xA, yA) lies on S1 + λ S2 = 0 ∀ λ ∈ R

Similarly point B(xB, yB) lies on S1 + λ S2 = 0 ∀ λ ∈ R

∴ S1 + λ S2 = 0 is the family of circles through the intersection of S1 = 0 and S2 = 0.

hope it helps u !!1

Answer 2

Answer:

We know that

x² + y² +2gx +2fy + c = 0 is an equation of a circle with centre at ( - g, - f) & radius √(g²+f²-c)

So For circle S1 :

Centre = C1 = (1,3)

Radius =R1 = √(1²+3²-9) = 1

So For circle S2 :

Centre = C2 = (0,-1)

Radius =R1 = √(0²+1+8) = 3

So C1C2

= Distance between centres

= √ ( 1- 0) ² + ( 3 +1) ²

= √17

So

C1C2+R1+R2 = 4+√17

Hence the answer is (A) 4+√17

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