To determine the distance between the points [tex]\((a \cos \theta + b \sin \theta, 0)\)[/tex] and [tex]\((0, a \sin \theta - b \cos \theta)\)[/tex], we can follow these steps:
1. Identify the coordinates: The coordinates of the first point are [tex]\((x_1, y_1) = (a \cos \theta + b \sin \theta, 0)\)[/tex]. The coordinates of the second point are [tex]\((x_2, y_2) = (0, a \sin \theta - b \cos \theta)\)[/tex].
2. Apply the distance formula: The distance [tex]\(d\)[/tex] between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] in the Cartesian plane is given by:
[tex]\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\][/tex]
3. Substitute the coordinates into the distance formula:
[tex]\[
d = \sqrt{(0 - (a \cos \theta + b \sin \theta))^2 + ((a \sin \theta - b \cos \theta) - 0)^2}
\][/tex]
4. Simplify the expressions inside the square root:
[tex]\[
d = \sqrt{(-(a \cos \theta + b \sin \theta))^2 + (a \sin \theta - b \cos \theta)^2}
\][/tex]
5. Square each term:
[tex]\[
d = \sqrt{(a \cos \theta + b \sin \theta)^2 + (a \sin \theta - b \cos \theta)^2}
\][/tex]
6. Expand the squares:
[tex]\[
d = \sqrt{(a \cos \theta)^2 + 2(a \cos \theta)(b \sin \theta) + (b \sin \theta)^2 + (a \sin \theta)^2 - 2(a \sin \theta)(b \cos \theta) + (b \cos \theta)^2}
\][/tex]
7. Combine like terms:
Notice that some of the mixed terms cancel out:
[tex]\[
2(a \cos \theta)(b \sin \theta) - 2(a \sin \theta)(b \cos \theta) = 0
\][/tex]
So, we have:
[tex]\[
d = \sqrt{(a \cos \theta)^2 + (b \sin \theta)^2 + (a \sin \theta)^2 + (b \cos \theta)^2}
\][/tex]
8. Factorize the expression:
[tex]\[
d = \sqrt{a^2 (\cos^2 \theta + \sin^2 \theta) + b^2 (\sin^2 \theta + \cos^2 \theta)}
\][/tex]
Since [tex]\(\cos^2 \theta + \sin^2 \theta = 1\)[/tex], this simplifies to:
[tex]\[
d = \sqrt{a^2 + b^2}
\][/tex]
Therefore, the correct answer is:
[tex]\[
\boxed{\sqrt{a^2 + b^2}}
\][/tex]
So, the correct choice is (c) [tex]\(\sqrt{a^2 + b^2}\)[/tex].
