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10-1.Circle and System of Circles
hard
A circle $S$ passes through the point $(0,1)$ and is orthogonal to the circles $(x-1)^2+y^2=16$ and $x^2+y^2=1$. 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 $(-8,1)$
A
$(B,D)$
B
$(B,C)$
C
$(A,C)$
D
$(A,D)$
(IIT-2014)
Solution
Let the cirlce be
$x^2+y^2+2 g x+2 f y+c=0$ $\quad\quad……….(1)$
given circles
$x ^2+ y ^2-2 x -15=0 $ $\quad\quad……….(2)$
$x ^2+ y ^2-1=0$ $\quad\quad……….(3)$
$(1)$ and $(2)$ are orthogonal
$\Rightarrow \quad-g+0=\frac{c-15}{2} $
$\Rightarrow \quad 0+0=\frac{c-1}{2} $
$\Rightarrow \quad c=1 \& g=7$
so the cirle is $x^2+y^2+14 x+2 f y+1=0 \quad$ it passes thrgouh
$(0,1) \Rightarrow \quad 0+1+0+2 f+1=0 $
$ \Rightarrow \quad x^2+y^2+14 x-2 y+1=0$
Centre $(-7,1)$
$\text { radius }=7$
Standard 11
Mathematics
