Write the distance between the points

Question:

Write the distance between the points A (10 cos θ, 0) and B (0, 10 sin θ).

Solution:

We have to find the distance between $\mathrm{A}(10 \cos \theta, 0)$ and $\mathrm{B}(0,10 \sin \theta)$.

In general, the distance between $\mathrm{A}\left(x_{1}, y_{1}\right)$ and $\mathrm{B}\left(x_{2}, y_{2}\right)$ is given by,

$\mathrm{AB}=\sqrt{\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}}$

So,

$\mathrm{AB}=\sqrt{(10 \cos \theta-0)^{2}+(0-10 \sin \theta)^{2}}$

$=\sqrt{10^{2}\left(\sin ^{2} \theta+\cos ^{2} \theta\right)}$

But according to the trigonometric identity,

$\sin ^{2} \theta+\cos ^{2} \theta=1$

Therefore,

$\mathrm{AB}=10$

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