First slide

Equation of line in Straight lines

Question

Let n be the number of points having rational coordinates at a fixed distance from the point $(0, \sqrt{3})$ . Then

Moderate

A

$n > 2$
B

$n \leq 1$
C

$n \leq 2$
D

$n = 1$

Solution

Let $A (x_{1}, y_{1}) and B (x_{2}, y_{2})$ be two points at equal distance a from $P (0, \sqrt{3}) .$ Then
$x_{1}^{2} + {(y_{1} - \sqrt{3})}^{2} = a^{2}$
and $x_{2}^{2} + {(y_{2} - \sqrt{3})}^{2} = a^{2}$
Subtracting (2) from (1), we get
$x_{1}^{2} - x_{2}^{2} + y_{1}^{2} - y_{2}^{2} - 2 \sqrt{3} (y_{1} - y_{2}) = 0$
Equating irrational and rational parts to zero, we get
$y_{1} = y_{2}$
and $x_{1}^{2} - x_{2}^{2} + y_{1}^{2} - y_{2}^{2} = 0$
Thus, $y_{1} = y_{2} = a (say)$
and $x_{1}^{2} - x_{2}^{2} = b^{2} (say)$
Hence, at most two such points are possible whose coordinates are (b,a) and (-b,a). In a special case, when b=0, only one such point is possible.

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