Distance Between Points Calculation

To find the distance between two points $P_1 = (x_1, y_1)$ and $P_2 = (x_2, y_2)$, we use the distance formula:

$ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} $

In this case, the points are:

• $P_1 = (a \cos \alpha, 0)$
• $P_2 = (0, a \sin \alpha)$

Let's substitute the coordinates into the distance formula:

$ d = \sqrt{(0 - a \cos \alpha)^2 + (a \sin \alpha - 0)^2} $

Simplify the terms inside the square root:

$ d = \sqrt{(-a \cos \alpha)^2 + (a \sin \alpha)^2} $

$ d = \sqrt{a^2 \cos^2 \alpha + a^2 \sin^2 \alpha} $

Factor out $a^2$:

$ d = \sqrt{a^2 (\cos^2 \alpha + \sin^2 \alpha)} $

Using the trigonometric identity $\cos^2 \alpha + \sin^2 \alpha = 1$:

$ d = \sqrt{a^2 \times 1} $

$ d = \sqrt{a^2} $

The square root of $a^2$ is the absolute value of $a$.

$ d = |a| $

Therefore, the distance between the two given points is $|a|$.