Find the length of the straight line joining the pairs of points whose polar coordinates are $(a, \frac{π}{2})$ and $(3 a, \frac{π}{6})$

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Solution

(i) $(a, \frac{π}{2})$
Then, $r = a$ and $θ = \frac{π}{2}$
$x = r c o s (\frac{π}{2}) = 0$
$y = r s i n (\frac{π}{2}) = a s i n (\frac{π}{2}) = a$
$(x_{1}, y_{1}) = (0, a)$
(ii) $(3 a, \frac{π}{6})$
Then, $r = 3 a$ and $θ = \frac{π}{6}$
$x = r c o s (\frac{π}{6}) = 3 a \times \frac{\sqrt{3}}{2}$
$y = r s i n (\frac{π}{2}) = 3 a s i n (\frac{π}{6}) = \frac{3 a}{2}$
$(x_{2}, y_{2}) = (\frac{30 \sqrt{3}}{2}, \frac{3 a}{2})$
Now applying distance formula,

$d = \sqrt{(x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}}$

$d = \sqrt{{(0 - \frac{3 \sqrt{3}}{2} a)}^{2} + {(a - \frac{3 a}{2})}^{2}}$
$\Rightarrow d = a \sqrt{\frac{28}{4}}$
$\Rightarrow d = a \sqrt{7}$

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