Maximize a value for a semicircle of given radius
We are given a semi circle with radius R. We can take any point on the circumference let it be P.Now, from that point P draw two lines to the two sides of diameter. Let the lines be PQ and PS.
The task is to find the maximum value of expression PS2 + PQ for a given R.
Input : R = 1 Output : 4.25 (4*1^2 + 0.25) = 4.25 Input : R = 2 Output : 16.25 (4 * 2^2 + 0.25)= 16.25
Let F = PS^2 + PQ. We know QS = 2R (Diameter of the semicircle)
-> We also know that triangle PQS will always be right angled triangle irrespective of the position of point P on the circumference of circle
1.)QS^2 = PQ^2 + PS^2 (Pythagorean Theorem) 2.) Adding and Subtracting PQ on the RHS QS^2 = PQ^2 + PS^2 + PQ - PQ 3.) Since QS = 2R 4*R^2 + PQ - PQ^2 = PS^2 + PQ => 4*R^2 + PQ - PQ^2 = F 4.) Using the concept of maxima and minima differentiating F with respect to PQ and equating it to 0 to get the point of maxima for F i.e. 1 - 2 * PQ = 0 => PQ = 0.5 5.) Now F will be maximum at F = 4*R^2 + 0.25