How do you do this question?

18π

Step-by-step explanation:

Arc length for a parametric equation is:

s = ∫ₐᵇ √((dx/dt)² + (dy/dt)²) dt

dx/dt = -9 sin(9t)

dy/dt = 9 cos(9t)

s = ∫₀²ᵖⁱ √((-9 sin(9t))² + (9 cos(9t))²) dt

s = ∫₀²ᵖⁱ √(81 sin²(9t) + 81 cos²(9t)) dt

s = ∫₀²ᵖⁱ 9 dt

s = 9t |₀²ᵖⁱ

s = 18π

You can check your answer with geometry.  The curve is a circle centered at the origin with radius 1.  From 0 to 2π, the particle makes 9 revolutions, or travels 18π radians.  So the arc length is 1 × 18π = 18π.

step-by-step explanation:

points a and b are equidistant from the desired line, so the line you find by using them will go through point r.

using points other than a and b will give the desired line only if they are equidistant from r. if they are randomly located, the perpendicular line you create will not necessarily go through point r, hence will not be a solution to the problem.