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Find the coordinates of a point on the line $x + y = 5$ which is equidistant from $(6, 4)$ and $(5, 2)$.
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Sol. Let the required point be $(x, y)$ which is equidistant from $(6, 4)$ and $(5, 2)$ (1 Mark)
$\therefore (6 – x)^2 + (4 – y)^2 = (5 – x)^2 + (2 – y)^2$
$\Rightarrow 2x + 4y = 23$ --- (i) (1/2 Mark)
Since point $(x, y)$ also lies on the line $x + y = 5$
$\therefore x + y = 5$ --- (ii)
Solving (i) and (ii), we get $x = -\frac{3}{2}$ and $y = \frac{13}{2}$ (1/2 Mark)
So, the coordinates of the required point are $(-\frac{3}{2}, \frac{13}{2})$
$\therefore (6 – x)^2 + (4 – y)^2 = (5 – x)^2 + (2 – y)^2$
$\Rightarrow 2x + 4y = 23$ --- (i) (1/2 Mark)
Since point $(x, y)$ also lies on the line $x + y = 5$
$\therefore x + y = 5$ --- (ii)
Solving (i) and (ii), we get $x = -\frac{3}{2}$ and $y = \frac{13}{2}$ (1/2 Mark)
So, the coordinates of the required point are $(-\frac{3}{2}, \frac{13}{2})$