A circle S passes through the point (0, 1) and is orthogonal to the circles (𝑥−1)2+𝑦2=16 and 𝑥2+𝑦2=1. Then
(A) Radius of S is 8 (B) Radius of S is 7
(C)Centre of S is (-7, 1) (D)Centre of S is (-8. 1)
Explanation
Step 1: Point (0, 1) se guzarne ki condition
Sawaal ke anusar, circle (0,1) se guzarta hai, toh:
02+12+2g(0)+2f(1)+c=0
Step 2: Pehle circle ke saath Orthogonality
Pehla circle hai: (x−1)2+y2=16⟹x2+y2−2x−15=0
Yahan g1=−1,f1=0,c1=−15.
Orthogonality ki condition 2g1g2+2f1f2=c1+c2 ka use karte hue:
2(−1)(g)+2(0)(f)=−15+c
−2g=−15+c
Step 3: Dusre circle ke saath Orthogonality
Dusra circle hai: x2+y2=1⟹x2+y2−1=0
Yahan g2=0,f2=0,c2=−1.
Condition apply karne par:
2(0)(g)+2(0)(f)=−1+c
0=−1+c
Step 4: g aur f ki values nikalna
Equation 3 se humein mila c=1. Ab ise baki equations mein rakhte hain:
Step 5: Centre aur Radius
Circle S ka centre (−g,−f) hota hai:
Ab radius (r) nikalte hain:
r=g2+f2−c
r=72+(−1)2−1
r=49+1−1=49
Right Answer
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