Question

Find the length of the straight line joining the pairs of points whose polar coordinates are $(-3,{45}^o)$ and $(7,{105}^o)$

Answer 1

Given polar co-ordinates

(i) $A(-3,{45}^{\circ })$

Then, $r=-3$ and $\theta ={45}^{\circ }$

$x=rcos\theta =-3cos{45}^{\circ }=\frac{-6}{2\sqrt{2}}$

$y=rsin\theta =-3sin{45}^{\circ }=\frac{-6}{2\sqrt{2}}$

$A(x_1,y_1)=(\frac{-3}{\sqrt{2}},-\frac{3}{\sqrt{2}})$

(ii) $B(7,{105}^{\circ })$

Then, $r=7$ and $\theta ={105}^{\circ }$

$x=rcos\theta =7cos{105}^{\circ }=7\times \frac{1-\sqrt{3}}{2\sqrt{2}}$

$y=rsin\theta =7sin{105}^{\circ }=7\times \frac{1+\sqrt{3}}{2\sqrt{2}}$

$B(x_2,y_2)(\frac{7-7\sqrt{3}}{2\sqrt{2}},\frac{7+7\sqrt{3}}{2\sqrt{2}})$

Now applying distance formula to calculate $AB=\sqrt{(x_1-x_2{)}^2+(y_1-y_2{)}^2}$

$AB=\sqrt{{(\frac{-6}{2\sqrt{2}}-\frac{7-7\sqrt{3}}{2\sqrt{2}})}^2+{(\frac{-6}{2\sqrt{2}}-\frac{7+7\sqrt{3}}{2\sqrt{2}})}^2}$

$AB=\sqrt{79}$