First slide

Cartesian plane

Question

If points A(3,5) and B are equidistant from $H (\sqrt{2}, \sqrt{5})$ and B has rational coordinates, then AB=

Moderate

A

$\sqrt{7}$
B

$\sqrt{{(3 - \sqrt{2})}^{2} + {(5 - \sqrt{5})}^{2}}$
C

$\sqrt{34}$
D

None of these

Solution

Let $B \equiv (α, β), where α, β$ are rational.
Given ${HA}^{2} = {HB}^{2}$
$\Rightarrow {(3 - \sqrt{2})}^{2} + {(5 - \sqrt{5})}^{2} = {(α - \sqrt{2})}^{2} + {(β - \sqrt{5})}^{2}$
$\Rightarrow 41 - 6 \sqrt{2} - 10 \sqrt{5} = α^{2} + β^{2} + 7 - 2 \sqrt{2} α - 2 \sqrt{5} β$
$\Rightarrow α^{2} + β^{2} - 34 + 2 (3 - α) \sqrt{2} + 2 (5 - β) \sqrt{5} = 0$
Since $α and β$ are rational,
$\begin{array}{l} 5 - β = 0 \Rightarrow β = 5 \\ and 3 - α = 0 \Rightarrow α = 3 \end{array}$
$∴ AB = 0$

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