A variable line ax + by + c = 0, where a, b, | vedclass

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Question

$a x+b y+c=0$

$a+c=2 b \Rightarrow a-2 b+c=0$

${x=1, y=-2} $

${(1,-2)=(\alpha, \beta)}$

$(x-1)^{2}+(y+2)^{2}=\gamma$

$ \Rightarrow {x^2

Vedclass · Accepted Answer

$7$

Vedclass · Answer

$a x+b y+c=0$

$a+c=2 b \Rightarrow a-2 b+c=0$

${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 $x^{2}+y^{2}-4 x-4 y-1=0$

$\Rightarrow 4-8=5-\gamma-1$

$\gamma=8$

$\alpha+\beta+\gamma=1-2+8=7$

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

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