First slide

Equation of line in Straight lines

Question

If every point on the line $(a_{1} - a_{2}) x + (b_{1} - b_{2}) y = c$ is equidistant from the points $(a_{1}, b_{1})$ and $(a_{2}, b_{2})$ then $2 c =$

Moderate

A

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

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

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

none of these

Solution

Let $(h, k)$ be any point on the given line then
$\begin{array}{l} n {(h - a_{1})}^{2} + {(k - b_{1})}^{2} = {(h - a_{2})}^{2} + {(k - b_{2})}^{2} \\ \Rightarrow 2 (a_{1} - a_{2}) h + 2 (b_{1} - b_{2}) k = a_{1}^{2} + b_{1}^{2} - a_{2}^{2} - b_{2}^{2} \\ \Rightarrow (a_{1} - a_{2}) h + (b_{1} - b_{2}) k \end{array}$
$= (1 / 2) (a_{1}^{2} + b_{1}^{2} - a_{2}^{2} - b_{2}^{2})$ (i)
Since $(h, k)$ lies on the given line
$(a_{1} - a_{2}) h + (b_{1} - b_{2}) k = c$ (ii)
Comparing (i) and (ii) we get $c = (1 / 2) (a_{1}^{2} + b_{1}^{2} - a_{2}^{2} - b_{2}^{2}) .$

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