- Home
- Standard 11
- Mathematics
10-1.Circle and System of Circles
normal
A variable line $ax + by + c = 0$, where $a, b, c$ are in $A.P.$, is normal to a circle $(x - \alpha)^2 + (y - \beta)^2 = \gamma$ , which is orthogonal to circle $x^2 + y^2- 4x- 4y-1 = 0$. The value of $\alpha + \beta + \gamma$ is equal to
A
$3$
B
$5$
C
$10$
D
$7$
Solution
$a x+b y+c=0$
$a+c=2 b \Rightarrow a-2 b+c=0$
${x=1, y=-2} $
${(1,-2)=(\alpha, \beta)}$
$(x-1)^{2}+(y+2)^{2}=\gamma$
$ \Rightarrow {x^2} + {y^2} – 2x + 4y + 5 – \gamma = 0$
it is orthogonal to $x^{2}+y^{2}-4 x-4 y-1=0$
$\Rightarrow 4-8=5-\gamma-1$
$\gamma=8$
$\alpha+\beta+\gamma=1-2+8=7$
Standard 11
Mathematics
