$B$ and $C$ are two points on the circle $x^{2} + y^{2} = a^{2} .$ From a point $A (b, c)$ on that circle $A B = A C = d .$ The equation to $B C$ is

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$b x + a y = a^{2} - d^{2}$

$b x + a y = d^{2} - a^{2}$

$b x + c y = 2 a^{2} - d^{2}$

$2 (b x + c y) = 2 a^{2} - d^{2}$

answer is D.

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Detailed Solution

$A (b, c)$ is a point on the circle $\begin{aligned} x^{2} + y^{2} = a^{2} \\ \Rightarrow b^{2} + c^{2} = a^{2} . . . . . . . (1) \end{aligned}$
Equation of the circle with centre $A (b, c)$ and radius $^{'} d^{'}$ is
$\begin{aligned} (x - b)^{2} + (y - c)^{2} = d^{2} \\ \Rightarrow x^{2} + y^{2} - 2 b x - 2 c y + a^{2} - d^{2} = 0 \end{aligned}$
$∴ R \cdot A of x^{2} + y^{2} = a^{2}$ and
$x^{2} + y^{2} - 2 b x - 2 c y + a^{2} - d^{2} = 0 is$
$\Rightarrow 2 (b x + c y) = 2 a^{2} - d^{2}$