If the equation $(a_{1} - a_{2}) x + (b_{1} - b_{2}) y = c$ represents the perpendicular bisector of the segment joining $(a_{1}, b_{1}) a n d (a_{2}, b_{2})$ then 2c=

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$a_{1} a^{2} + b_{1} b^{2}$

${a_{1}}^{2} + {b_{1}}^{2} + {a_{2}}^{2} + {b_{2}}^{2}$

${a_{1}}^{2} + {b_{1}}^{2} - {a_{2}}^{2} - {b_{2}}^{2}$

${a_{1}}^{2} - {b_{1}}^{2} + {a_{2}}^{2} - {b_{2}}^{2}$

answer is A.

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Detailed Solution

$l e t A (a_{1}, b_{1}) a n d B (a_{2}, b_{2})$

$N o w m i d p o i n t o f A B = (\frac{a_{1} + a_{2}}{2}, \frac{b_{1} + b_{2}}{2})$

If the equation $(a_{1} - a_{2}) x + (b_{1} - b_{2}) y = c$ represents the perpendicular bisector of the segment joining $(a_{1}, b_{1}) a n d (a_{2}, b_{2})$ then

mid point lies on the above line

$\Rightarrow (a_{1} - a_{2}) (\frac{a_{1} + a_{2}}{2}) + (b_{1} - b_{2}) (\frac{b_{1} + b_{2}}{2}) = c \Rightarrow {a_{1}}^{2} + {b_{1}}^{2} - {a_{2}}^{2} - {b_{2}}^{2} = 2 c$