If OA is the initial line and A the end of the first revolut

If OA is the initial line and A the end of the first revolution of the spiral. and if the tangent to the spiral at A is drawn then the perpendicular to OA at O will meet the tangent at some point B. Establish that the length of the segment OB is equal to the circumference of the circle with radius OA; hence, the area of MOB is equal to the area of this circle. [Hint: The slope of the tangent at A is 2 pi.

Solution

Slope of the tangent = 2pi gives

tan OAB = 2pi

OR OB/OA =2pi

Or OB = 2pi (r) where r is the radius of the circle

Area of triangle AOB = 1/2 base (height) = 1/2 OB (OA)

0.5(2pir) r = pir² = Area of circle

Hence proved.